Final answer:
Claudius becomes visibly uncomfortable and anxious during the play that Hamlet stages to reflect the real circumstances of King Hamlet's death, thus betraying his guilt.
Step-by-step explanation:
How Claudius reacts to the play within the play, which mirrors the true circumstances of King Hamlet's death, reveals much about his character and guilt. During the performance of The Murder of Gonzago, which Hamlet has staged to “catch the conscience of the King,” Claudius becomes visibly uncomfortable and anxious.
This reaction occurs when the scene closely resembling the method in which Claudius purportedly murdered King Hamlet is acted out.
His distress is so acute that he rises and calls for lights to end the play abruptly, thereby betraying his guilt. This pivotal scene underscores the theme of appearance versus reality, as well as the complexity of action and inaction within the play.