Before I attempt to explain the meaning of "born again," I would like to address what Jesus meant by "see the kingdom of God." The Greek εἴδω (eidō), which is here translated "see," is translated as such 314 times. The same Greek word is also translated "know" 281 times in the New Testament. In other words, εἴδω means to "experience something for oneself, to recognize and understand it." If Nicodemus was to ever to recognize the King of the Messianic Kingdom, he needed to be born again.
The phrase "born again," is translated from two Greek words that mean "born," γεννάω (gennaō), and "again," ἄνωθεν (anōthen), however, Strong's Concordance, a book known for its circular reasoning, suggests that ἄνωθεν means "from above." It bases that on the context of this passage, and not upon the Greek, itself. Circular reasoning!
In reality, "again" means "something repeated." I often carries with it the idea of a teacher telling his piano student to "play it again." It implies an improvement over the first attempt. The same goes for one being "born again." We are born the first time, physically, out of a water-filled womb, into a world no purer than we are. The new birth, or second birth, is a spiritual birth. Instead of being born into a physical realm, the new birth results in a life into the spiritual realm. Look at the passage, and see if this is not clearly taught!
The physical birth is from one's mother (v. 4), while the spiritual birth is from the miraculous work of the Holy Spirit (v. 5-8). These verses say, "Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. Marvel not that I said unto thee, Ye must be born again. 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."
We know the King, because the Holy Spirit has given us eyes to recognize Him!