What affect does the Player's speech about Hecuba have on Hamlet?
Hamlet feels inadequate and frustrated with his own lack of action. The Player is able to generate and convey passion and emotion in his speech about Hecuba's grief over the death of Priam, yet this situation is not a real one; the Player is just acting. Hamlet, on the other hand, has real cause to feel grief and to act, yet he has done nothing. He asks what would the Player do "Had he the motive and the cue for passion/That I have?" So he questions himself: "Am I a coward?" And he declares that so far all he done to achieve the vengeance the Ghost wants is to use words.
Ah, but words! That fact gives him the idea of using the play-within-a-play to reenact the murder scene and watch Claudius' response. If the king even flinches, Hamlet will know that Claudius is guilty and thus have the evidence he's been seeking that the Ghost speaks truth. Hamlet will use words "to catch the conscience of the king" and lay a trap for Claudius.
Hamlet is surprised that the Player can produce such emotions over something that has not really happened. He chides himself for experiencing his father's death and mother's remarriage and not doing anything. He says, "What's Hecuba to him, or he to Hecuba, / That he should weep for her?" In other words, the Player cried over an imaginary person he can't know, yet Hamlet can't convince himself to take action for someone he loved.