When we first meet Samson, his life is a total bummer: he's blind, he's in prison, his family and friends have abandoned him, and he even feels abandoned by his God. His down-and-out situation becomes only sadder as he and some visitors retell just how awesome Samson's life used to be as a hero of Israel divinely blessed and chosen by God. We know this is the exposition because it fills us in on the backstory and sets us up to expect something to change—stat.
What action? Samson is rather famous for lacking in traditional action, since Samson's physical circumstances remain the same for the majority of the play. But there is plenty of spiritual and psychological action. It may not make great TV, but it does make good poetry: as Samson continues to complain about his life and berate himself for having gotten himself in this situation, we realize that even though Samson seemed to have a lot going for him pre-haircut and loss of strength/freedom/eyesight, things weren't all perfectly rosy before.
When his wife Dalila appears, we know for sure that Samson made a lot of poor choices. So we're getting set up for the final action to be something that redeems his character.
After some depressing conversations with Samson's father and former wife, Samson's captors order him to make a public appearance in their theater to entertain everyone with some feats of strength. Samson flatly refuses, but minutes later has some kind of inner revelation which tells him that agreeing to appear is part of God's plan. So, off he goes! Why is this the climax? Because it's the point at which everything changes.
Just like Greek and Roman tragedy, the biggest moment of action happens off-stage. Like Samson's father and the Chorus, we only know about it because of a messenger. After hearing a horrible cry, a messenger runs onto the scene and explains that Samson used his incredible strength to collapse the theater, killing all of his enemies and himself.
Talk about "falling action."
The bad: Samson is dead. The good: he totally died a hero. Considering how miserable he was as a prisoner, Manoa and the Chorus agree that his death is probably a good thing. He's free from his prison; God has apparently forgiven him; and, oh yeah, everything is now all nice and peaceful with all the Philistine fat cats dead. Thanks, Samson!