I think the obvious connection for the audience to make would be Gertrude's guilt, not Claudius'. They have no clue that regicide has been committed. But they do know that Gertrude married very quickly after the king's death, as did the woman in the play. I think that would have been the source for any discomfort for the audience, and there's Hamlet rubbing it in, asking his mother if she likes the play, will the Player Queen keep her word not to remarry.

Could the audience take that as the reason Claudius is upset? The connection between the play and Gertrude and Hamlet's taunting of her is upsetting Claudius on her behalf, and so they get up and leave.