This term originates from a passage in John 3, where Jesus tells a Pharisee named Nicodemus that people can't enter heaven without first being born anothen—that is, born from above. However, in addition to "from above," the Greek word anothen can also mean "again," and this is the sense in which Nicodemus understands Jesus' words. The fact that Nicodemus mistakenly thinks that Jesus means "born again" explains his response:
"How can a man be born when he is old? Can he enter a second time into his mother's womb and be born?"Had Nicodemus realized that Jesus meant people must be "born from above," he wouldn't have responded this way. He would have understood that Jesus meant a spiritual rebirth rather than a physical one. So what's the problem? Why is this misunderstanding such a big deal? Simple: anothen is a Greek word, but Jesus and Nicodemus would have been speaking Aramaic. The "again/from above" double meaning doesn't exist in Aramaic, so the sequence of events described in John simply would not have happened.
Could it be that Jesus actually said "born again" in Aramaic, and the writer of John just happened to use a word that also meant "from above"? Nope. The other three times he uses anothen—including once just a few verses later—the "from above" meaning is unambiguous. And Thayer's Lexicon says the word is "often used of things which come from heaven, or from God as dwelling in heaven," which suits this context too perfectly to be a coincidence. So if this conversation took place, Jesus would have used an Aramaic equivalent of "born from above."
So how did Nicodemus manage to misinterpret his words so spectacularly? Well, he didn't. The author of John fabricated the story, not realizing that the double meaning would be impossible in Jesus' native tongue. According to the NET Bible, "the author uses the technique of the 'misunderstood question' often to bring out a particularly important point: Jesus says something which is misunderstood by...someone else, which then gives Jesus the opportunity to explain more fully and in more detail what he really meant."
To resolve this problem, apologists must argue not only that this well-educated Pharisee was an idiot who couldn't understand that being "born from above" would be a spiritual birth and not a physical reemergence from the womb, but that the translation into Greek created the "again/from above" double meaning purely by chance. I have an alternative for them: stop struggling. Relax. Let it go. Realize that your mental acrobatics are futile, and accept that the Bible is not a reliable record of Jesus' life—or of most other things, for that matter. It may be upsetting at first, but once you've unchained yourself from this ancient book for a while, you'll probably feel a lot better. At least, I do.