The mob dispersed after Scout's conversation with Mr. Cunningham because it ceased to be just a mob. It became a collection of individuals, each of whom was recognizable.
Mr. Cunningham's safety was in anonymity. To the mob mentality, an action taken by a mob is less reprehensible than the same act taken by an individual. Thus, when Scout treated Mr. Cunningham as an individual, she destroyed the mob. No one person in it was then safe from being taken out of the mob and being made a unique person.
Mr. Cunningham was identified as a family man, a father. This was quite different from what he was doing in connection with the mob. Personal diginity and virtue can be more easily set aside within the anonymity of a mob. These were brought back to Mr. Cunningham by Scout.
This riveting scene in Chapter 15 is developed through strong dramatic irony. We know that the group of men who confront Atticus at the jail is a lynch mob that has come for Tom Robinson. Scout, however, does not understand the danger of the situation as it unfolds. When she innocently places herself in the middle of it, she sees not a group of potential murderers but a neighbor she recognizes, Walter Cunningham. Scout talks to Mr. Cunningham as the family friend she believes him to be, persisting until he must finally acknowledge her. Scout reminds Mr. Cunningham of who he really is--a father and someone whom Atticus has helped. When he begins to think as an individual, Cunningham himself disperses the mob. The facelessness and lack of personal responsibility that make up the psychology of a mob is destroyed by Scout's personal interaction with Cunningham. Atticus later explains the nature of mob psychology to his children:
A mob's always made up of people, no matter what. Mr. Cunningham was part of a mob last night, but he was still a man . . . . you children last night made Walter Cunningham stand in my shoes for a minute. That was enough.
Because of how kind The Finch's had been to Mr.Cunningham's son.