What does it tell us when Petruchio turns up at his wedding dressed like a slob?
The part that you are talking about happens in Act III, Scene 2.
I think that the reason Petruchio does this is to "tame" Kate. Up until this point, he has tried to tame her by being nice. But now he is going to completely turn around and use a different tactic. He is now going to try to humiliate her. I think he believes that this will show her that she is not as valuable as she thinks.
So that is why he shows up to the wedding dressed like a bum -- he wants to show her that he does not really care about marrying her -- it's not a big deal to him.
There are two ways of interpreting this.
Either Petruchio can be seen as a slob, clearing showing no effort being put into his attire and thus having little care about the wedding itself. Or, as P states later in the scene, he is proving a point that looks do not matter to him, for she is marrying him for himself not his clothes, and hence teaching Katherina not to judge a book by it's cover.