It is true that Dill comes to stay with the Finches because he says he is being mistreated by his mother who has a new man in her life, Dill's new step-father. However, there is more to this story than meets the eye. The truth is that Dill feels neglected and discriminated against. He has been cast aside because of his age. Another example of ageism in the book. Much like the racism, sexism, and other types of discrimination in the book, everyone has a 'cross to bear' so to speak. In Dill's case his is that he seems to be invisible to the most important adults in his life, his mother and now even his new step-father. A man who has made promises to Dill, but has failed to keep them. In this regard, Dill is a lot like Boo Radley, unseen and unheard, almost forgotten. This brings to light the reasons for Dill's keen interest in Boo Radley throughout the book.
Well, he says he ran away (and came there) because his new father had chained him in the basement, but the reality is much more common, and in a way sadder (even if less dramatic): his parents seem to get along better without him, and he's happier with the Finches.
Dill stays over the Finche's because he had run away from home. He thought that his mother and "new" father were not paying attention to him.