Why did Tom help Mayella with her chores?

Because he felt sorry for her.

Tom helps Mayella for two reasons. First, Mayella asks him to help her. This is important, because back then it would have been unheard of for him to not help her. As a black man it was important that he respond appropriately to the request of anyone white. The other reason he helps her is revealed at the trial when he says he "felt sorry for her". This creates a big problem for Tom as it would be impudent of him to feel sorry for a white woman because she is better than him simply because she is white; thus there should be no reason to feel sorry for her.

He feels bad for Mayella so he does the chores since they are so poor