Short answer
Because the Prophet Muhammad (ﷺ) forbade them to men, and a believer trusts that God (﷾) knows best (33:36). Gold and silk are not evil. Women wear them, and the Qur'an promises gold and silk to the believers in Paradise (22:23). So a Muslim man gives up a small luxury now for a far greater one to come. Islam does not deny men beauty: they have fine clothes and perfume, and the Prophet (ﷺ) wore a silver ring (Bukhari 5866). The rule comes from his own words. Holding silk in his right hand and gold in his left, he said that both are forbidden to the men of his community (Abu Dawud 4057), and lawful for its women (Nasa'i 5148). The New Testament, too, tells believing women not to make gold their adornment (1 Timothy 2:9).
Full explanation
Silk in one hand, gold in the other
Ali ibn Abi Talib saw the Prophet (ﷺ) take silk in his right hand and gold in his left. He said: both of these are forbidden to the males of my community (Abu Dawud 4057). Abu Musa reports the other half: gold and silk are lawful for the women of the community and forbidden to its men (Nasa'i 5148). Al-Tirmidhi records the same words. He notes that many other Companions report this teaching, among them Umar, Ali, Anas and Hudhayfa (Tirmidhi 1720). The Prophet (ﷺ) also forbade eating and drinking from vessels of gold and silver, and wearing silk or sitting on it (Bukhari 5837).
Ali ibn Abi Talib reports that the Prophet (ﷺ) took silk and held it in his right hand, and took gold and held it in his left, and said: both of these are forbidden to the males of my community.
Abu Musa reports that the Prophet (ﷺ) said: gold and silk have been made lawful for the women of my community and forbidden to its men.
Abu Musa al-Ash'ari reports that the Prophet (ﷺ) said: wearing silk and gold has been made unlawful for the males of my community and lawful for its females. Al-Tirmidhi adds that the same is reported from Umar, Ali, Uqba ibn Amir, Anas, Umm Hani, Hudhayfa and others, and grades it hasan sahih.
Hudhayfa reports that the Prophet (ﷺ) forbade them to drink or eat from vessels of gold and silver, and to wear silk and silk brocade or sit on them.
The ring he threw away
The Prophet (ﷺ) once wore a ring, and the people began to wear gold rings like it. When he saw them, he threw his own away and said: I will never wear it. Then he wore a silver ring, and the people wore silver rings (Bukhari 5866). Another time he saw a gold ring on a man's hand, pulled it off and threw it down. He said the man was putting a live coal from Hell on his hand. When the Prophet (ﷺ) had gone, people told the man to pick the ring up and make some use of it. He said: no, by Allah (﷾), I will never take it when the Messenger of Allah (ﷺ) has thrown it away (Muslim 2090).
Ibn Umar reports that the Prophet (ﷺ) wore a ring, and the people began to wear gold rings like it. When he saw them, he threw his own away and said: I will never wear it. Then he wore a silver ring, and the people wore silver rings.
Ibn Abbas reports that the Prophet (ﷺ) saw a gold ring on a man's hand, pulled it off and threw it away, saying: one of you wants a live coal from Hell and puts it on his hand. When the Prophet (ﷺ) had left, the man was told to take his ring and make use of it. He said: no, by Allah, I will never take it when the Messenger of Allah (ﷺ) has thrown it away.
Abu Hurayra reports that the Prophet (ﷺ) forbade the wearing of a gold ring.
Not evil, but kept for Paradise
Gold and silk are not unclean. Muslim women wear both. When the Prophet (ﷺ) was given silk garments, he sent them to his Companions to cut into head coverings for their wives (Muslim 2068c). The Qur'an promises both to the believers in Paradise: "they shall be adorned therein with bracelets of gold and pearls; and their garments there will be of silk" (22:23; also 35:33 and 18:31). The Prophet (ﷺ) said that whoever wears silk in this world will not wear it in the Hereafter (Bukhari 5832). So a Muslim man gives up a small luxury now for a far greater one to come.
Ibn Umar reports that the Prophet (ﷺ) said: whoever wears silk in this world has no share in the Hereafter. Later, when silk garments were given to him, he sent one each to Umar, Usama ibn Zayd and Ali, saying: tear them and make head coverings for your women. When Umar asked about it, he said he had not sent it for him to wear, but so that he might make use of it.
“Allah will admit those who believe and work righteous deeds, to Gardens beneath which rivers flow: they shall be adorned therein with bracelets of gold and pearls; and their garments there will be of silk.”
“Gardens of Eternity will they enter: therein will they be adorned with bracelets of gold and pearls; and their garments there will be of silk.”
“For them will be Gardens of Eternity; beneath them rivers will flow; they will be adorned therein with bracelets of gold, and they will wear green garments of fine silk and heavy brocade: They will recline therein on raised thrones. How good the recompense! How beautiful a couch to recline on!”
Anas ibn Malik reports that the Prophet (ﷺ) said: whoever wears silk in this world will not wear it in the Hereafter.
Keeping a rule without being told why
The Prophet (ﷺ) did not explain the ban, and a Muslim does not need him to. The Qur'an says: "It is not fitting for a Believer, man or woman, when a matter has been decided by Allah and His Messenger to have any option about their decision" (33:36). The man who would not pick up his ring understood this. He did not ask why; it was enough for him that the Prophet (ﷺ) had thrown it away (Muslim 2090). For a Muslim, trusting that God (﷾) knows best is part of faith itself. And Islam does not forbid beauty: "Say: Who hath forbidden the beautiful (gifts) of Allah, which He hath produced for His servants ...?" (7:32). Men have their own fine clothes, perfume and silver.
“It is not fitting for a Believer, man or woman, when a matter has been decided by Allah and His Messenger to have any option about their decision: if any one disobeys Allah and His Messenger, he is indeed on a clearly wrong Path.”
Ibn Abbas reports that the Prophet (ﷺ) saw a gold ring on a man's hand, pulled it off and threw it away, saying: one of you wants a live coal from Hell and puts it on his hand. When the Prophet (ﷺ) had left, the man was told to take his ring and make use of it. He said: no, by Allah, I will never take it when the Messenger of Allah (ﷺ) has thrown it away.
“Say: Who hath forbidden the beautiful (gifts) of Allah, which He hath produced for His servants, and the things, clean and pure, (which He hath provided) for sustenance? Say: They are, in the life of this world, for those who believe, (and) purely for them on the Day of Judgment. Thus do We explain the signs in detail for those who understand.”
What a Muslim man may wear
A silver ring, as the Prophet (ﷺ) and his Companions wore (Bukhari 5866). A trim of silk about two fingers wide: Umar wrote to the Muslims that the Prophet (ﷺ) forbade silk "except this much", and showed it with his two fingers (Bukhari 5829). And silk for a medical need. The Prophet (ﷺ) let Abd al-Rahman ibn Awf and al-Zubayr wear silk shirts for a skin disease that made them itch (Bukhari 2919). The scholars agree that the gold ring itself is forbidden to men, as al-Nawawi records. So the rule is narrow, and it leaves a man much that is fine to wear.
Ibn Umar reports that the Prophet (ﷺ) wore a ring, and the people began to wear gold rings like it. When he saw them, he threw his own away and said: I will never wear it. Then he wore a silver ring, and the people wore silver rings.
Abu Uthman reports that while they were in Adharbayjan, Umar wrote to them that the Prophet (ﷺ) forbade wearing silk except this much, and showed it with his two fingers, the index and the middle finger.
Anas reports that the Prophet (ﷺ) allowed Abd al-Rahman ibn Awf and al-Zubayr to wear silk shirts because they had a skin disease that made them itch.
It says that a gold ring is forbidden to men by the consensus of the scholars, because of the authentic hadith that forbid gold to the men of this community, and quotes al-Nawawi that the scholars agree gold is forbidden to men.
The Bible also warns against gold
Islam is not alone in this. The New Testament tells believing women not to make gold their adornment: "not with broided hair, or gold, or pearls, or costly array" (1 Timothy 2:9). It also says: "Whose adorning let it not be that outward adorning of plaiting the hair, and of wearing of gold" (1 Peter 3:3). The Torah draws a line between what men and women wear: "neither shall a man put on a woman’s garment" (Deuteronomy 22:5). In all three, what a believer wears is a matter for God (﷾).
“In like manner also, that women adorn themselves in modest apparel, with shamefacedness and sobriety; not with broided hair, or gold, or pearls, or costly array;”
“Whose adorning let it not be that outward adorning of plaiting the hair, and of wearing of gold, or of putting on of apparel;”
“The woman shall not wear that which pertaineth unto a man, neither shall a man put on a woman’s garment: for all that do so are abomination unto the Lord thy God.”
Perspectives
The rule is arbitrary
Gold and silk harm no one. Forbidding them to men alone is arbitrary and unequal.
Gold and silk harm no one, so forbidding them is arbitrary.
Rebuttal
Islam does not say they harm anyone: women wear them, and the Qur'an promises both in Paradise (22:23). A Muslim man gives them up because God (﷾) and His Messenger (ﷺ) decided it (33:36). A believer keeps any command of God (﷾) in the same way.
Allowing them to women and not to men is unequal.
Rebuttal
Different is not unequal. Islam gives men and women some different rules, in dress among them, and the same reward: "for them has Allah prepared forgiveness and great reward" (33:35). The Torah, too, keeps some clothing to each (Deuteronomy 22:5).
The Prophet (ﷺ) himself wore a gold ring.
Rebuttal
He did, before the ruling. When the people copied him, he threw it away, said he would never wear it, and wore silver (Bukhari 5866). He kept the rule himself first.
The Islamic perspective
Gold and silk are kept for women in this life, and for every believer in Paradise. A Muslim man gives them up here because the Prophet (ﷺ) asked it, trusting God's (﷾) wisdom. Islam does not deny him beauty: a silver ring, fine clothes and perfume are his.
The Prophet (ﷺ) forbade gold and silk to the men of his community and allowed them to its women.
The scholars agree that a gold ring is forbidden to men.
Gold and silk are promised to the believers in Paradise.
A silver ring, a trim two fingers wide and silk for a medical need are allowed to men.