As I read the whole message that Jesus gave to Nicodemus, I noticed a footnote from verse 11a that reads "The Greek construction suggest that verses 11-21 contain a direct quotation. This testimony of Jesus was given to a member of the Sandedrin." That made me think more about those verses as one speech rather than individual verses that are really good. It is a beautiful testimony of Jesus.
Verse 14 says the son of man must be lifted up. That is exactly the same testimony that Jesus gives of himself in 3 Nephi 21:13-19. Just as Jesus told Nicodemus that a man can't enter into the kingdom of God without being born of the spirit, Jesus told the Nephites that they couldn't enter into his rest if they were not clean. Only those who could enter his rest were those "who have washed their garments in [his] blood because of their faith, and the repentance of all their sins, and their faithfulness unto the end" (3 Ne 27:19).
I gained some additional insights as I cross referenced to D&C 84:24. This verse taught about the Israelites who couldn't see God because "they hardened their hearts and could not endure his presence (italics added)." What a great perspective. It's not that God is mean and chooses not to let us enter into his Kingdom if we aren't baptized by fire. It's a natural consequence of our decisions! If we are unclean, we cannot be in his kingdom because we ourselves could not withstand His glory.
The way to be able to endure his presence is by being reborn of the spirit. By letting the Holy Ghost change our very beings until we become like God. We need the Holy Ghost in our lives -- every single day. As Alma asked, "if ye have experienced a change of heart, and if ye have felt to sing the song of redeeming love...can ye feel so now?"