What Scout's words to Mr. Cunningham do is they remind him that he is an individual person.  This makes him stop thinking like a member of a mob and more like an actual person.

Psychologists say that when you are part of a mob you lose your inhibitions.  You don't really feel like you are a person and you don't feel like you have to obey the same rules that regular people do.

But Scout makes him realize that he is a person and that the rules still apply to him.  This makes him rethink lynching Tom Robinson and he gets his friends to stop too.

Scout began talking about his son Walter, a friend of hers at school. She also talked about the entailment he had. She had no idea really what entailment was, but she knew it was bad. It's really not so much what she said but how she said it that made the difference. She talked to Mr. Cunningham as if the two of them were buddies.

No one else around the circle really talked. When Scout was done, she said to say hey to Walter for her. Mr. Cunningham grew some humanity in that moment. She humbled him. He told the boys to clear out and that talk from that little girl stifled and prevented a possible altercation between the men and Atticus, or the men and Tom Robinson. She has no idea that she did it.