After Roger tries to steal her purse, Mrs. Jones tells Roger that she was young once and wanted things she could not get. This surprises Roger. He realizes that she understands his longing better than he thought.
When Roger tries to rob Mrs. Jones, he has no idea what he is in for. He seems completely baffled by getting caught, answering her questions in short, sullen bursts that are borderline respectful. He does answer her questions though, and he does decide to stay instead of running.
You thought I was going to say, but I didn’t snatch people’s pocketbooks. Well, I wasn’t going to say that." Pause. Silence. "I have done things, too, which I would not tell you, son—neither tell God, if he didn’t already know. (p. 3)
She does not reveal to him what she had done, but she makes it clear that it was comparable or worse to stealing an old lady’s purse.
Roger realizes that what he did was wrong, but it does not make him a bad person. His desperation got the better of him. He makes an effort to be good from then on, and stay where she can see him. He wants to be trusted by her.