It's about sacrifice, and the power of belief. For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.

Our standing with God depends, not on our race, social standing, abilities, or outward appearance, but on our exercising faith in Jesus’ ransom sacrifice. His greatest manifestation of love, the most loving thing that a parent can do, he did for mankind. That was the giving of the life of his own faithful, only-begotten Son.

It was popularized as a standard bible quote for purposes of distancing a church body from Calvinistic roots.

I think it's because it's used to remind people of the promise of resurrection and eternal life, if they have faith in God and Jesus Christ.

"John 3:16 (chapter 3, verse 16 of the Gospel of John) is one of the most widely quoted verses from the Christian Bible. It has been called the "Bible in a nutshell" because it is considered a summary of some of the most central doctrines of traditional Christianity: For God so loved the world, that He gave His only begotten Son, that whosoever believeth in Him shall not perish, but have everlasting life. – John 3:16 (KJV) A typical interpretation of the verse might be as follows:[original research?] For God so loved the world... — God is a god of love and this love motivates his action in the rest of the verse ...that He gave... — God giving something to the world of humanity that he owned, his son, as a sacrifice ...His only begotten[1] Son... — the firstborn of all creation Jesus of Nazareth is also the Son of God. ...that whosoever... — that any person ...believeth... — being saved is based on faith and trust in Jesus to save or faith ...in Him... — the belief being in Jesus, God's means of salvation for humanity ...shall not perish... — implies the fate of those who do not believe: eternal death ...but have everlasting life. — shows the reward of those who believe: eternal life" "The verse occurs in a narrative in the third chapter of John taking place in Jerusalem. Nicodemus, a member of the ruling council, comes to talk with Jesus, whom he calls Rabbi. Jesus' "miraculous signs" have convinced Nicodemus that Jesus is "from God". In reply, Jesus declares, "I tell you the truth, no one can see the kingdom of God unless he is born again" (verses 5–6). John 3:16 summarizes Jesus' lesson to Nicodemus: that belief in Jesus is the path to eternal life. In this passage, Jesus proclaims himself the Messiah and lays out important aspects of Christian theology. This shows a contrast with the Gospel of Mark, for instance, in which Jesus often tries to keep his divinity secret until his final trip to Jerusalem." Source and further information: http://en.wikipedia.org/wiki/John_3:16

...because a lot of people like its message and have shared it with others over the centuries?

John 3:16 has over the years been made famous by those spreading Christianity. The verse has for a long time been translated as “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” – that’s how it appears in the KJV. Because it has become famous it has attracted more attention from those who like to look deeper. They tell us that the idea of God Almighty literally begetting a son was not present during or soon after the time of Jesus and that it was not a central part of the teachings of Jesus or the belief of the early church. They point out that of the four Gospels only in John’s account is Jesus referred to as the “only BEGOTTEN Son” of God. If it had meant anything the other three Gospel writers would certainly not have wanted to leave it out. By comparison all Gospel writers made sure that they told us that Jesus rode a donkey into Jerusalem, a mere detail compared to Jesus being the “only BEGOTTEN Son” of God. The word begotten has connotations of impregnation through sexual intercourse or insemination. And even if one wanted to shut out those associations with that word when used in such holy context, involving God Almighty as it does, the Church makes sure with its creedal formula that you don’t. It emphasises that Jesus was “begotten NOT made” forcing you to think of the unthinkable. Perhaps it is for these and other scholarly reasons that the eminent Christian theologians who revised the KJV have removed the word BEGOTTEN (perhaps as an addition meant to align the Gospel with Paul's teachings) and rendered the same verse in the Revised Standard Version as: “For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.” This of course still leaves questions about the way in which God came to have a “one and only son”, but at least it cleans Him of the animal process of begetting, and gets a step closer to agreeing with the HQ which declares in HQ 112:3 that “He begetteth not, nor is He begotten” There are those who give the OT, being a Hebrew scripture, a purely Jewish interpretation. It then throws a lifeline to those who want to differ from the unholy “BEGOTTEN” view and want the Bible itself to support them. Psalm 2:7 quotes God Himself using the word when saying to David on his becoming a king “Thou art my Son; this day have I BEGOTTEN thee”. This is very clear evidence that the word can be used as a metaphor. However, those who look at Psalm 2:7 in that light have to contend with Paul’s claim, in Acts 13:33, that Psalm 2:7 is referring to the coming of Jesus.

Because people like it and what it says.