Wherein I'll catch the conscience of the King.
-- Hamlet, 2.2.531-2
The play to which Hamlet refers to is "The Mousetrap." A play about murder and a situation similar to that of Hamlet's father -- or so Hamlet believes.
He stages the play with the intent of getting definitive proof against his uncle Claudius. Though Hamlet already believes that his uncle is guilty, he hesitates in taking action. He hangs back, waiting for the right moment to carry through his revenge.
Thus conscience does make cowards of us all,
And thus the native hue of resolution
Is sicklied o'er with the pale cast of thought.
(3.1.82-4)
The arrival of the traveling players marks the perfect opportunity for Hamlet to test his uncle. If he lacked proof before, if he had feared that his personal feelings were perhaps clouding his judgment, the king's own performance during "The Mousetrap" could very well hold the key. And so, once the players arrive, Hamlet quickly puts into motion a "trap" to catch the king, under the guise of simply having the players perform a selected work.
For only Hamlet does the play hold a deeper significance, a hidden meaning that he hopes Claudius will also recognize. If his uncle does, then he will have effectively taken the bait and Hamlet would have his proof, his confirmation that his father's ghost was genuine. Any hesitation Hamlet may have felt before would likely be removed, given the success of his test. His trap.
The plan? Hamlet (along with the help of loyal friend, Horatio) would observe the king as "The Mousetrap" was performed, and see how the words, the action, and the story affected him. This of course would be predicated on the belief that, although Claudius was capable of murder and total treachery, he would still have a conscience that could be affected by what he saw (together with his own memory).
Ophelia: The king rises.
Hamlet: What, frighted with false fire?
Queen Gertrude [to Claudius]: How fares my lord?
Polonius: Give o'er the play.
King Claudius: Give me some light. Away.
Courtiers: Lights, lights, lights!
(3.2.243-8)
And so, the play does the trick.
To those who are aware of the situation, Claudius's reaction confirms every suspicion. As Hamlet so eloquently noted, "What, frighted with false fire?" Indeed, Claudius is trapped by his own actions. There was no actual accusation made, not directly at least. But the fact that he was so disturbed by what he saw could only mean that he, as Hamlet had hoped, understood the subtle message being sent. To Claudius, it was equivalent to someone calling him out, exposing him as the murderer he is in front of the world. His secret, no more.
