It's probably the most memorized Bible verse, simply because it sums up the Gospel. God loves us so much that He sent Jesus, and whoever believes in Jesus will go to heaven.
Jesus said the words in this verse to Nicodemus, a Pharisee that confessed Jesus is a teacher from God. But he didn't understand what Jesus was saying about being born again. He thought Jesus was talking about being born again like a baby, but that's not it at all. Jesus is talking about being born again in the Spirit. Jesus wants us to have a change of heart. He wants us to live in truth and in the light, and we can begin doing that by believing in Jesus. See, Nicodemus admitted Jesus is a teacher from God. But Jesus is not just a teacher, He's the son of God.
Our Preschool class memorizes John 3:16 because it reminds us of the love of God and who Jesus is.
Watch this very moving short film about the story of love and sacrifice.