Hamlet's "revision" of the play is intended to hurt his mother for what he perceives as infidelity and to reveal Claudius as a murderer, at least to him.
His mother would have been most affected by the conversation between the Player King and the Player Queen, particularly when she declares over and over again that she would never remarry if he should die.
Of course, much of this play takes place in pantomime. As a result, what happens after the Player King begins to sleep is merely action, not words. However, these actions hit their intended targets hard! First Lucianus, the nephew of the Player King and supposed to be representing Claudius, pours poison in the Player King's ear. Then he woos the Player Queen (Gertrude) into his arms.
These actions cause great distress in Claudius and Gertrude, with Claudius commanding that the play be stopped. Hamlet did get his answer.