What does John 8:58 mean?

In the gospel of John, there are seven instances where Jesus makes a statement alluding to His own divinity. These are known as the seven "I AM" statements. When Moses asked God how He should be identified to Israel, God said "I AM WHO I AM…Say this to the people of Israel: 'I AM has sent me to you.'" This phrase implies the simplest expression of God's nature: He just is, He must be. When Jesus uses this phrasing, He is deliberately invoking that same essence. Earlier in this very debate, Jesus made the second of these statements, claiming "I am the light of the world" (John 8:12).

Now, Jesus is responding to disbelief from His critics. Jesus indicated that Abraham looked forward to the coming of the Messiah, and "rejoiced" to see Jesus' day finally arrive. Jesus word choices, in the original Greek, imply a kind of knowledge which is inborn, innate, and natural. This led His critics to jeer: "you're not even fifty, how can you have seen Abraham?"

Jesus answers with what some refer to as the "hidden 'I AM'" statement, since it's not typically counted among the others. Ironically, this is perhaps the most direct of all of Jesus' uses of this phrase. The meaning is certainly not lost on His critics. When Jesus claims the title of "I AM," everyone listening knows exactly what He means: that He is God (John 10:33). Those who suggest that Jesus never actually claimed to be God should consider the reaction of the religious leaders to His words, shown in the next verse. They are so enraged that they attempt to stone Jesus (John 8:59) right then and there.