Short answer
He meant being born "from above", that is, from God (﷾). The Greek word in John 3:3 is anothen. It can mean "again", but its basic meaning is "from above". Elsewhere John uses the word three more times, and each time it means from above. Even the King James Version translates it "from above" in John 3:31. Nicodemus took it to mean a second birth from his mother, and Jesus corrected him: this birth is of the Spirit. By the second century, churches had tied the verse to baptism. Jesus himself said plainly what leads into life. When a man asked how to gain eternal life, he said, "keep the commandments". That is the way Islam teaches: obey God (﷾).
Full explanation
The word John uses
The Greek word in John 3:3 and 3:7 is anothen: ano means above and the ending -then means from. The King James Version translates it "again" in those two verses. John uses the same word three more times. In 3:31 the KJV has "he that cometh from above". In 19:11 it has power "given thee from above". In 19:23 it has a coat "woven from the top". The New Revised Standard Version translates 3:3 itself as born "from above".
“Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God.”
“He that cometh from above is above all: he that is of the earth is earthly, and speaketh of the earth: he that cometh from heaven is above all.”
“Jesus answered, Thou couldest have no power at all against me, except it were given thee from above: therefore he that delivered me unto thee hath the greater sin.”
“Then the soldiers, when they had crucified Jesus, took his garments, and made four parts, to every soldier a part; and also his coat: now the coat was without seam, woven from the top throughout.”
Translates John 3:3 as seeing the kingdom of God only by being born "from above", and gives "born anew" as the alternative in a footnote.
How Christians read it
Nicodemus hears the word as "a second time", and Jesus answers that a person must be born of water and of the Spirit. Christians read this as the new birth the Holy Spirit gives. Catholic and Orthodox teaching links it to baptism through the words "born of water". Many evangelicals speak of being born again at the moment of personal faith in Christ.
“Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother’s womb, and be born?”
“Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.”
Teaches that baptism is the washing of regeneration, which brings about the birth of water and the Spirit without which, citing John 3:5, no one can enter the kingdom of God.
How the verse came to mean baptism
By the middle of the second century, Justin Martyr, a Christian teacher in Rome, was quoting the saying as "born again" to explain baptism. In baptism, converts were said to be regenerated in water. Catholic teaching today still calls baptism the birth of water and the Spirit.
Describing Christian baptism in Rome in the middle of the second century, Justin says converts are brought to water and there "regenerated", and he supports this with Christ's saying that unless a person is born again he will not enter the kingdom of heaven.
Teaches that baptism is the washing of regeneration, which brings about the birth of water and the Spirit without which, citing John 3:5, no one can enter the kingdom of God.
A birth God (﷾) gives, not a rite
Jesus explains the birth he means with the wind, which blows where it wills. It is something God (﷾) does, which no one can arrange. Read as "from above", the saying speaks of God's (﷾) work in a person. It is not a ceremony or a single moment of decision.
“The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit.”
“He that cometh from above is above all: he that is of the earth is earthly, and speaketh of the earth: he that cometh from heaven is above all.”
What Jesus said leads into life
When a man asked Jesus directly what to do for eternal life, he said to keep the commandments. He said the kingdom belongs to those who do the will of his Father. He also said that whoever keeps and teaches the commandments will be called great. Islam teaches the same obedience to God (﷾). The Qur'an says Jesus confirmed the Law and made lawful some of what was forbidden before. Only God (﷾) changes His Law, through a prophet.
“And, behold, one came and said unto him, Good Master, what good thing shall I do, that I may have eternal life? And he said unto him, Why callest thou me good? there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments.”
“Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.”
“Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.”
“"'(I have come to you), to attest the Law which was before me. And to make lawful to you part of what was (Before) forbidden to you; I have come to you with a Sign from your Lord. So fear Allah, and obey me.”
“"Those who follow the messenger, the unlettered Prophet, whom they find mentioned in their own (scriptures),- in the law and the Gospel;- for he commands them what is just and forbids them what is evil; he allows them as lawful what is good (and pure) and prohibits them from what is bad (and impure); He releases them from their heavy burdens and from the yokes that are upon them. So it is those who believe in him, honour him, help him, and follow the light which is sent down with him,- it is they who will prosper."”
Verse by verse
John 3:5 — Gospel of John, chapter 3, verse 5
“Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.”
Cited for
"Born of water and of the Spirit" points to baptism: the new birth happens in the water of baptism, as Catholic teaching states.
The response
In the next breath Jesus explains this birth with the wind, which blows where it wills and cannot be traced. That is God's (﷾) own doing, not a rite a person performs. And the word John uses in 3:3 is the word he uses for what comes from above.
Where it lands — The saying never names baptism. Jesus explains this birth with the wind, which no one controls, and John's own word for it means from above. It is a birth God (﷾) gives, not a rite a person performs.
John 3:16 — Gospel of John, chapter 3, verse 16
“For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.”
Cited for
In the same chapter Jesus says that whoever believes in him will not perish but have everlasting life. Belief, not keeping the commandments, is the way into life.
The response
Belief in God's (﷾) messenger is part of Islam too, but belief was never set against obedience. John records Jesus saying that eternal life is to know "the only true God, and Jesus Christ, whom thou hast sent". When a man asked Jesus directly what to do for eternal life, he said to keep the commandments. He also said that not everyone who calls him Lord will enter the kingdom, but only the one who does the will of his Father.
Where it lands — Belief and obedience belong together. When Jesus was asked the way to eternal life, the answer he gave was to keep the commandments.
Perspectives
Mainstream Christian
Being born again is the new birth the Holy Spirit gives, without which no one sees God's kingdom. Churches differ on whether it comes through baptism or through personal faith.
Jesus says a person must be born of water and of the Spirit to enter the kingdom of God.
Rebuttal
John records these words, and the saying never names baptism. Jesus explains the birth at once with the wind, which blows where it wills. It is God's (﷾) doing, not a rite.
Catholic teaching calls baptism the birth of water and the Spirit.
Rebuttal
That is the Catholic Church's teaching today. But the saying itself never names baptism. Jesus explains the birth with the wind, which no one can arrange.
John intends both meanings, "again" and "from above", since Nicodemus hears "a second time".
Rebuttal
Nicodemus did hear "a second time", and Jesus corrected him: this birth is of the Spirit, not of the womb. Everywhere else John uses this word it means from above, and the New Revised Standard Version translates 3:3 that way.
The Islamic perspective
Jesus spoke of a birth from above: God's (﷾) work in a person, not a ritual. The way into life he gave was obedience to God's (﷾) commandments, which is what Islam teaches.
Outside the born-again saying (3:3 and 3:7), John uses the word three times. Each time it means from above or from the top.
The New Revised Standard Version translates John 3:3 as born from above.
By the second century the saying was being quoted as "born again" to explain baptism.
Asked how to gain eternal life, Jesus said to keep the commandments.