This is a great question, and you're not the first person to have it.
Today, the phrase "born again" can have any one of a host of meanings. It can have meaning in the secular world, such as a "born again" politician who changes political parties, or in the religious world, where "born again Christian" is sometimes used to differentiate one from a "regular" Christian.
And in fact, the original hearer apparently had this same question as you, as you see in verse 4:
Nicodemus said to him, “How can a man be born when he is old? Can he
enter a second time into his mother's womb and be born?” (John 3:4
Before jumping into what Jesus's intention was, it may be helpful to take a step back to verse 2, to see how Nicodemus approaches Jesus.
This man came to Jesus by night and said to him, “Rabbi, we know that you are a teacher come from God, for no one can do these signs that you do unless God is with him.”
(John 3:2 ESV)
Nicodemus, a member of the Jewish ruling council (v1), was a man in a position of authority. As a ruler of the Jews and teacher of Israel (v10), he is coming from a position in which he considers a teacher like Jesus, and evaluates him. "We know..." he says, from a position of authority. By default this puts Jesus at an inferior position.
But this is not how man must come to God--he, the creator, is the judge, and man, the creature, is the one being judged. In fact Jesus will, in a few verses, answer Nicodemus in the same way that Nicodemus has approached him. Just as Nicodemus came as if speaking on behalf of the council saying "we know..." Jesus responds in v11 saying "we speak of what we know, and bear witness to what we have seen" in an almost tongue-in-cheek way of saying "we know a few things too, we do".
But at present, Jesus turns Nicodemus around, saying, in essence "You think you see what God is doing? Let me tell you, you don't have a clue. In fact, you couldn't see that unless..." and then there's that tricky phrase. Jesus tells Nicodemus that the only way to see what God is doing is by being "born again".
Nicodemus apparently misunderstands this (v2), and so Jesus rephrases the assertion in v5, stating that he must be born of water and spirit. As argued elsewhere, this phrase ("born of water and spirit") is simply another way of putting "born again" in terms that Nicodemus ought to understand--that he must have a purifying and transforming spiritual rebirth.
But while v5 is put in more spiritual terms, v3 is a bit more practical. It is a metaphorical way of saying that we are so weighed down by sin that we are blinded to what God is working for us. What can't see the kingdom unless we first shed all of our sin. But even if we were to never sin again, we have every past sin to weigh us down as well. In fact, I suspect that Nicodemus is not totally lost on the metaphor, but is continuing the dialog using Jesus's metaphorical language--saying in order to shed our history of sin, we would need to go back into our mothers' wombs to start over.
But, while he may have understood the metaphor, he ought to have seen that there is a way to shed all of our sin, that we can be made clean. As Jesus points out in v10, Nicodemus should have known this because he is the teacher of Israel. This means that he was well-acquainted with what we know as the Old Testament, where these themes are present (again, I'd refer you to this answer for more on that background.