Answer: What Hamlet means when he says "The play's the thing/Wherein I'll catch the conscience of the King" is that Claudius will give himself away and reveal his guilt when he sees the play that Hamlet has been working on.
Step-by-step explanation: In Hamlet, Hamlet decides to direct a play in which a nephew kills his uncle and King in order for him to confirm that Claudius was the one who murdered his father. In that way, as Hamlet believes that his uncle's guilty conscience will betray him, the Prince tells his friend Horatio to pay attention to Claudius' gestures and suspicious behaviour during the play.