Jem, Dill, and Scout come even though they have strict instructions not to follow him.  In this case, however, it is a good thing for Atticus and Tom that the kids did what they weren't supposed to do.  Scout talks with one of the men--Mr. Walter Cunningham-- in the mob about his son...the two of them are school friends.  The very conversation reminds the man that they are all living people and that Mr. Cunningham owes Atticus Finch for past favors.  They are all connected and her speaking of Walter's son to him at this moment in time softens his heart.  He orders the mob to leave the area, and no one is hurt.  Scout doesn't realize how tense the situation could have been and how much danger she and her family was in, but Atticus and Tom do. 

Recalling the mob scene of Chapter XXII of The Adventures ofHuckleberry Finn, in which Sheburne confronts a mob--

The pitifulest thing out is a mob; that's what an army is--a mob; they don't fight with courage that's born in them, but with courage that's borrowed from their mass and from their officers. But a mob without any man at the head of it, is beneath pitifulness--

In To Kill a Mockingbird, the lynch-party are the "Old Sarum" bunch. "an enormous and confusing tribe" from the northern part of the county. Like the mob in Twain's novel, they, too, are lacking in courage individually, but they possess it as a group. However, it is little Scout who turns it into "the pitifulest thing"; for, after she and Jem follow Atticus to the jailhouse, they see the old dusty cars pulling into the lot by the door where Atticus sits.

When one man puts his hands on Jem, Scout kicks him. As the tension mounts, and a man says, "All right, Mr. Finch, get  'em outa here," Scout recognizes Mr. Cunningham and addresses him, returning him to the office of leader of small mind. After they lose their leader, as suggested by Sheburne, the mob becomes the "pitifulest thing" so Mr. Cunningham tells them to follow him home.

It is the unexpected intervention of Scout that stops the lynching party in their tracks. She goes to the jail that night only because she is following Jem, who, being older and wiser, senses trouble. Atticus is at the jail that night as he is aware that an attempt might be made on Tom's life. With characteristic courage, he has gone there alone, but he ends up getting unexpected help from his children. Jem, too, shows courage in facing up to the men, but it is Scout who really saves the day - most unintentionally.

This episode is at once tense and comical. The comedy springs from the fact that Scout is so uncomprehending of what is going on, but the tension arises from the fact that there is very real threat from these men. However, they are nonplussed by the sudden appearance of an innocent little girl in their midst and her embarrassment at finding herself among a bunch of strangers. She tries to make connection with them, finally lighting upon Mr Cunningham, father of her classmate Walter, the only person she recognizes. Her innocent reaching out to him has the dramatic effect of defusing the hostility of the situation. He responds in kind, and the lynching party breaks up, without having menaced either Tom Robinson or his protector, Atticus. Scout therefore inadvertently saves her father, brother, and Tom. It is not until later that she realizes just what she has done, and then she is overwhelmed by the seriousness of the situation which at the time she was so unaware of: 'The full meaning of the night’s events hit me and I began crying' (chapter 16). 

We see in this episode, then, how even a mob set on murder might be disarmed. When Scout unwittingly appeals to Mr Cunningham's humanity, the men are no longer able to function as a mindlessly murderous group; they break up, and the danger is over.

In chapter 15, Atticus leaves his home after dinner to sit outside of the Maycomb jailhouse to prevent anyone from trying to harm his client, Tom Robinson, before the trial. The children follow Atticus and watch from nearby as a lynch mob arrives from the Meridian highway and surrounds him. Scout is extremely curious and runs out into the group of men, where she recognizes Mr. Cunningham. Scout is unaware of the gravity of the situation and attempts to make small talk with Mr. Cunningham. Scout brings up Mr. Cunningham's entailment and attempts to discuss his son, Walter Jr. After initially ignoring her, Mr. Cunningham finally acknowledges Scout, sympathizes with Atticus's difficult situation, and orders the lynch mob to leave.

Essentially, Scout's presence and her innocent questions influence Mr. Cunningham to stand in Atticus's shoes and view the situation from his perspective. Mr. Cunningham sympathizes with Atticus and realizes that he is putting him in a dangerous, precarious position. After viewing the situation from Atticus's point of view, Mr. Cunningham calls off the lynch mob. In chapter 16, Atticus teaches his children a lesson on mob mentality and acknowledges that it took a little girl to make Mr. Cunningham and the mob come to their senses. Atticus tells Jem and Scout,

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 . . . So it took an eight-year-old child to bring ‘em to their senses, didn’t it? . . . That proves something—that a gang of wild animals can be stopped, simply because they’re still human . . . you children last night made Walter Cunningham stand in my shoes for a minute. That was enough.

The person who really does the most to persuade the mob to back off is Scout, even though she does not know what she is doing.

What Scout does is to talk to Mr. Cunningham, thus singling him out as an individual and making him think as an individual.

Psychologists tell us that people behave one way when they are acting as indiviudals and another way when they feel like they are part of a faceless mob.  Cunningham was part of a faceless mob until Scout talked to him.  Once she did, he went back to being an individual.  He then realized what he was doing and backed off, taking the mob with him.