Short answer
Only good dreams. In Bukhari the Prophet (ﷺ) says that nothing is left of prophethood except the glad tidings. Asked what they are, he says: true good dreams. A good dream is one part in forty-six of prophethood, and any believer may have one. One part is not the whole. Allah (﷾) still guides, comforts and inspires His servants. He inspired the mother of Moses, and the Prophet (ﷺ) said that men were spoken to before this community without being prophets. None of this is prophethood: it brings no messenger, no new book and no one to obey as a prophet. In another report he said that messengership and prophethood have come to an end.
Full explanation
What the Prophet (ﷺ) said is left
In Bukhari's report he says that nothing is left of prophethood except the glad tidings. Asked what they are, he says: true good dreams. In Muslim's report he drew back the curtain of his room and saw the people praying behind Abu Bakr. He told them that nothing is left of the glad tidings of prophethood except the good dream that a Muslim sees or is shown. Read together, the two reports explain each other. What remains is the glad tidings that came with prophethood, not prophethood itself. In another report he first said that messengership and prophethood have come to an end, so there is no messenger and no prophet after him. When the people were troubled, he told them what remains, and named dreams.
Abu Hurayra reports that the Prophet (ﷺ) said nothing is left of prophethood except al-mubashshirat. Asked what they are, he said: true good dreams, which bring glad news.
Ibn Abbas reports that the Prophet (ﷺ) drew back the curtain of his room and saw the people praying in rows behind Abu Bakr. He told them that nothing is left of the glad tidings of prophethood (mubashshirat al-nubuwwa) except the good dream that a Muslim sees or that is shown to him.
Anas ibn Malik reports that the Prophet (ﷺ) said that messengership and prophethood have come to an end, so there will be no messenger after him and no prophet. The people were troubled, so he said: but there will be mubashshirat. He explained that these are the dreams of the Muslim, a portion of the portions of prophethood.
A part is not the whole
He said that a good dream is one of forty-six parts of prophethood. A part of a thing is not the thing. A drop is part of the sea, but nobody calls a drop the sea. And the dream is not kept for prophets. It is the one that a Muslim sees or is shown, so any believer may have one. It brings comfort or a warning. It does not bring a law, and nobody must obey the dreamer.
Anas ibn Malik reports that the Prophet (ﷺ) said a good dream of a righteous man is one of forty-six parts of prophethood.
Ibn Abbas reports that the Prophet (ﷺ) drew back the curtain of his room and saw the people praying in rows behind Abu Bakr. He told them that nothing is left of the glad tidings of prophethood (mubashshirat al-nubuwwa) except the good dream that a Muslim sees or that is shown to him.
Inspiration is not prophethood
The Qur'an says Allah (﷾) sent inspiration to the mother of Moses, and taught the bee. Neither is a prophet. The Prophet (ﷺ) said that among the Children of Israel there were men who were spoken to without being prophets. If anyone in his community is like them, he said, it is Umar. So being spoken to, inspired or shown glad news does not make someone a prophet. The door of Allah's (﷾) kindness stays open. Such inspiration binds no one. The sending of prophets has ended with the Seal of the Prophets.
“So We sent this inspiration to the mother of Moses: "Suckle (thy child), but when thou hast fears about him, cast him into the river, but fear not nor grieve: for We shall restore him to thee, and We shall make him one of Our messengers."”
“And thy Lord taught the Bee to build its cells in hills, on trees, and in (men's) habitations; Then to eat of all the produce (of the earth), and find with skill the spacious paths of its Lord: there issues from within their bodies a drink of varying colours, wherein is healing for men: verily in this is a Sign for those who give thought.”
Abu Hurayra reports that the Prophet (ﷺ) said that among the Children of Israel before them there were men who were spoken to without being prophets, and that if anyone in this community is like them, it is Umar.
“Muhammad is not the father of any of your men, but (he is) the Messenger of Allah, and the Seal of the Prophets: and Allah has full knowledge of all things.”
What the Ahmadi community says remains
The community reads the hadith differently. Its book on the finality of prophethood says the hadith does not mean that prophethood has ended. It means only that a ceiling has been placed on its size. So long as the glad tidings remain, it says, prophethood has not ended. Mirza Ghulam Ahmad wrote that the door of converse with Allah (﷾) will never be closed to the Prophet's (ﷺ) community until the Day of Judgment. He wrote that prophethood by way of reflection (zill), received purely through obedience to him, lasts until then. He also wrote that a prophet is one honoured with plentiful converse with Allah (﷾) and plentiful news of the unseen. He wrote that he alone of his community received so great a share.
A book of the Ahmadiyya community on the finality of prophethood. It reads the hadith "nothing remains of prophethood except mubashshirat" as meaning that prophethood has not ended but has had a ceiling placed on its magnitude, so long as mubashshirat remain.
Mirza Ghulam Ahmad writes that Muhammad (ﷺ) became Khatamul-Anbiya' (the Seal of the Prophets) in the sense that he possesses the seal without whose attestation no grace reaches anyone. He writes that independent prophethood has ended with the Prophet, but that prophethood by way of zill (reflection), received purely through obedience to him, will last until the Day of Judgment. He also writes that for his ummah the door of converse and communion with the Divine will never be closed until the Day of Judgment.
Mirza Ghulam Ahmad writes that by "prophet" he means only that he receives Allah's converse plentifully, and quotes the Mujaddid of Sirhind that one who receives such converse abundantly, with abundant news of the unseen, is called a prophet (p. 497). He writes that a prophet is one honoured with such plentiful converse with Allah, and such plentiful news of the unseen, as cannot be given to anyone except a prophet. He writes that he alone of his community received so great a share, and that for this reason he alone was given the title of Prophet (p. 498).
Why the hadith cannot be read that way
First, he named what is left and limited it: true dreams. In Muslim's report they are called the glad tidings of prophethood. Second, the report in Tirmidhi begins by saying that messengership and prophethood have ended. The exception that follows leaves only dreams. Third, Ibn Kathir gives Anas's report under 33:40 as proof that there is no prophet after him. Fourth, a dream that any Muslim may see cannot be the prophethood of someone sent to a people.
Ibn Abbas reports that the Prophet (ﷺ) drew back the curtain of his room and saw the people praying in rows behind Abu Bakr. He told them that nothing is left of the glad tidings of prophethood (mubashshirat al-nubuwwa) except the good dream that a Muslim sees or that is shown to him.
Abu Hurayra reports that the Prophet (ﷺ) said nothing is left of prophethood except al-mubashshirat. Asked what they are, he said: true good dreams, which bring glad news.
Anas ibn Malik reports that the Prophet (ﷺ) said that messengership and prophethood have come to an end, so there will be no messenger after him and no prophet. The people were troubled, so he said: but there will be mubashshirat. He explained that these are the dreams of the Muslim, a portion of the portions of prophethood.
Ibn Kathir explains that the verse forbids calling Zayd "son of Muhammad", since the Prophet (ﷺ) was not the father of any of the men. On "the Seal of the Prophets" he says the verse is explicit that there is no prophet after him. If there is no prophet after him, there is no messenger either, since every messenger is a prophet and not the other way round. He adds that the reports of this are mutawatir, from a group of Companions, and gives the hadith of the missing brick and the hadith of Anas.
Anas ibn Malik reports that the Prophet (ﷺ) said a good dream of a righteous man is one of forty-six parts of prophethood.
Much converse is not prophethood
The Ahmadi claim is that a follower to whom Allah (﷾) speaks often, and shows the unseen, may be called a prophet. The Prophet (ﷺ) said there were men before this community who were spoken to without being prophets. He said that if anyone in his community is like them, it is Umar. The Prophet (ﷺ) named Umar as the one most like them, and Umar was not a prophet. However often Allah (﷾) inspires a believer, it does not make him one. In Tirmidhi the Prophet (ﷺ) spoke of what is left after prophethood has ended, and named only the good dreams.
Abu Hurayra reports that the Prophet (ﷺ) said that among the Children of Israel before them there were men who were spoken to without being prophets, and that if anyone in this community is like them, it is Umar.
Anas ibn Malik reports that the Prophet (ﷺ) said that messengership and prophethood have come to an end, so there will be no messenger after him and no prophet. The people were troubled, so he said: but there will be mubashshirat. He explained that these are the dreams of the Muslim, a portion of the portions of prophethood.
Mirza Ghulam Ahmad writes that by "prophet" he means only that he receives Allah's converse plentifully, and quotes the Mujaddid of Sirhind that one who receives such converse abundantly, with abundant news of the unseen, is called a prophet (p. 497). He writes that a prophet is one honoured with such plentiful converse with Allah, and such plentiful news of the unseen, as cannot be given to anyone except a prophet. He writes that he alone of his community received so great a share, and that for this reason he alone was given the title of Prophet (p. 498).
Renewers, not prophets
The Prophet (ﷺ) also told his community what to expect. Allah (﷾) will raise for it, at the head of every hundred years, someone who renews its religion. That is a renewer, not a prophet. A renewer calls people back to the Book and the Sunnah. He brings no revelation that others must accept. In his farewell sermon the Prophet (ﷺ) left the Book of Allah (﷾) among them. He said that whoever holds fast to it will never go astray. Allah (﷾) has completed the religion (5:3) and promised to guard the Qur'an (15:9). The Ahmadi site argues that Muslims still need guidance. The Prophet (ﷺ) named the answer: the Book, and renewers who serve it.
Abu Hurayra reports that the Prophet (ﷺ) said Allah will raise for this community, at the head of every hundred years, someone who will renew its religion for it.
In Jabir's account of the farewell pilgrimage, the Prophet (ﷺ) said in his sermon: I have left among you the Book of Allah, and if you hold fast to it you will never go astray. He then asked the people what they would say about him, and they answered that he had conveyed the message, discharged his trust and given sincere counsel.
“This day have I perfected your religion for you, completed My favour upon you, and have chosen for you Islam as your religion.”
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
The community's page answers those who say no prophet is needed because the Qur'an and the Sunnah are preserved. It says Allah and the Prophet (ﷺ) do not agree, since the Prophet told us Jesus will come. It argues that Muslims still need guidance, and that the only way to bring them back to the right path is for Allah to send another prophet in Islam. It reads the hadith that Jesus will descend among Muslims with an imam from among them as a metaphor for the coming of Mirza Ghulam Ahmad.
Allah (﷾) is still close to every believer
Nothing here shuts the door on Allah (﷾). He says that when His servants ask about Him, He is close, and He answers the one who calls. He promises the obedient the company of the prophets. The Prophet (ﷺ) said that a true dream is glad news. What has ended is the sending of prophets. Allah's (﷾) nearness has not ended.
“When My servants ask thee concerning Me, I am indeed close (to them): I listen to the prayer of every suppliant when he calleth on Me: Let them also, with a will, Listen to My call, and believe in Me: That they may walk in the right way.”
“All who obey Allah and the messenger are in the company of those on whom is the Grace of Allah,- of the prophets (who teach), the sincere (lovers of Truth), the witnesses (who testify), and the Righteous (who do good): Ah! what a beautiful fellowship!”
Abu Hurayra reports that the Prophet (ﷺ) said nothing is left of prophethood except al-mubashshirat. Asked what they are, he said: true good dreams, which bring glad news.
Perspectives
The Ahmadi view
Ahmadis hold that prophethood has not wholly ended. They read the hadith on glad tidings as leaving a form of prophethood in the community. They teach that a prophet who follows Muhammad (ﷺ) can still come.
"Nothing remains of prophethood except glad tidings" means prophethood has not ended. It has a ceiling on its size.
Rebuttal
In Bukhari he names what is left and limits it to true dreams, which any Muslim may see. That is not the office of a prophet. In Muslim's report, what remains is the glad tidings of prophethood, one part in forty-six. In Tirmidhi he says in the same breath that there is no messenger and no prophet.
Prophethood by reflection (zill) continues, and the door of converse with Allah stays open.
Rebuttal
Every believer may call on Allah (﷾), and He answers (2:186). Inspiration may still come to whom He wills, but it binds no one and is weighed against the Qur'an and the Sunnah. Inspiration came to the mother of Moses, and to men before this community who were not prophets.
A follower to whom Allah speaks often, and shows the unseen, may be called a prophet.
Rebuttal
The Prophet (ﷺ) said there were men before this community who were spoken to without being prophets. He said that if anyone in his community is like them, it is Umar. The Prophet (ﷺ) named Umar as the one most like them, and Umar was not a prophet. However often Allah (﷾) inspires a believer, it does not make him one.
Muslims still need a prophet to guide them back to the right path.
Rebuttal
The Prophet (ﷺ) left the Book as the safeguard against going astray, and promised a renewer at the head of every century. A renewer calls people back to the Book and brings no new revelation. Allah (﷾) has completed the religion and promised to guard the Qur'an.
The Islamic perspective
Of prophethood only the glad tidings remain: the true dream. Inspiration, guidance and renewal go on, but prophethood and messengership have ended.
In Bukhari the Prophet (ﷺ) said nothing is left of prophethood except the glad tidings, which are true dreams. In Muslim he said nothing is left of the glad tidings of prophethood except the good dream.
He said messengership and prophethood have come to an end.
A good dream is one part in forty-six of prophethood, and any believer may see one.
Inspiration is not prophethood. Allah (﷾) inspired the mother of Moses, and men before this community were spoken to without being prophets.
Allah (﷾) renews the faith through renewers at the head of each century, and the Book is the safeguard.