Why did Jesus say "My God, my God, why have you forsaken me?"?

It's the first line of Psalm 22 [or 21 if you've got a Catholic Bible], and it may have been intended to refer to the whole Psalm.

That particular Psalm starts with expressions of anguish, despair, and the expectation of being killed. It gradually moves to expressions of hope for salvation, and of faith in God. It's also credited to David, believed to be an ancestor of Jesus. All these characteristics make it a suitable citation for the circumstances of the Crucifixion.

There are also explanations which focus on that single line of despair:

He passed in some superhuman manner through our human horror of pessimism. When the world shook and the sun was wiped out of heaven, it was not at the crucifixion, but at the cry from the cross: the cry which confessed that God was forsaken of God. And now let the revolutionists choose a creed from all the creeds and a god from all the gods of the world, carefully weighing all the gods of inevitable recurrence and of unalterable power. They will not find another god who has himself been in revolt. Nay (the matter grows too difficult for human speech), but let the atheists themselves choose a god. They will find only one divinity who ever uttered their isolation; only one religion in which God seemed for an instant to be an atheist.

-- G.K. Chesterton, "Orthodoxy"