Why did you think the boy decided not to flee?

Roger does not flee because he is moved by the kindness and caring nature of Mrs. Luella Bates Washington Jones, and he does not want her to think poorly of him.
When Mrs. Jones collars Roger and drags him up the street after putting him in a wrestling hold to stop him from stealing her purse, the boy is humiliated. She tells Roger he needs his face washed, and she drags him home with her. Once he and Mrs. Jones reach the boarding house, Roger hears voices in distant rooms. He realizes other people are living in this house. He does not try to run because if he does, he might experience even more embarrassment at the hands of the formidable Mrs. Jones. Then, as she prepares her supper, Mrs. Jones thinks aloud that Roger must be hungry to have snatched her purse from her. Roger tells her that he wanted some suede shoes.

“Well, you didn’t have to snatch my pocketbook to get some suede shoes,” said Mrs. Luella Bates Washington Jones. “You could of asked me.”

Roger is taken aback by this generous offer by Mrs. Jones after she has wrestled him in the street and dragged him by the collar. All he can utter is "M'am?" There is a long pause, and Roger thinks, "The door was open. He could make a dash for it down the hall. He could run, run, run, run, run!" Instead, he stays with the woman who does not criticize and who has so generously offered to share her supper, as well as buy him the shoes he wants.