Hamlet is intrigued by the fact that this actor, reciting a speech given to him by some playwright, or perhaps a speech he composed himself, is moved to weep upon the recitation. How can he be so moved by what is only an act? He pretends to weep for Hecuba and he's certainly never met her or been given any reason to feel emotion for her. Yet he can move himself to tears over her.
He compares this to himself, unable to take action despite a very real offense, the suspected murder of his father. Why is it that this actor can be moved so easily and he remains still?
His frustration leads him to the realization that perhaps he can use this power of the actors to his benefit, that he can use the play to show the king's guilt, that Claudius, if he sees his brother's murder reenacted will give some sign to show he did in fact do it.
In his soliloquy of Act II, Scene 2, Hamlet chastises himself for his weakness and inaction in avenging the murder of his father, and he considers a method to confirm the guilt of Claudius.
Steeped in melancholy over the death of his father and what he views as his mother's incestuous act of marriage to Claudius, Hamlet finds himself in a quagmire of thoughts and emotions that immobilize him. In his soliloquy of Act II, Scene 2, Hamlet muses upon the emotion that an actor of the visiting troupe brings forth in his speech about the Trojan queen Hecuba, the prototype for bereft and mourning women. He wonders what this man would do if he "[H]ad the motive and cue for passion" (Act II, Scene 2, line 517) that he has in his current situation.
Upon further introspection, Hamlet berates himself for his lack of passion and courage:
But I am pigeon-livered and lack gall
To make oppression bitter, or ere this
I should'a fatted all the region kites
With this slave's offal. Bloody, bawdy villain (Act II, Scene 2, lines 534-537)!
Further, Hamlet calls himself "an ass" for his inaction. He then remembers that sometimes people who watch a play whose plot resembles circumstances of their own lives are "struck so to the soul" (Act II, Scene 2, line 548) that they are driven to confess the crimes they have committed.
Resolved to act, Hamlet decides to have the actors perform a play whose plot involves situations similar to those which have recently occurred in reality. Then, as Claudius watches this play, Hamlet can "catch the conscience of the king" (Act II, Scene 2, line 562).
As with Hamlet's other soliloquies, his third soliloquy moves him toward action and provides more insights into his soul.