In "Thank you, M'am," why does Roger offer to go to the store?

Roger wants Mrs. Jones to trust him, and he wants to show his appreciation to her.

Mrs. Jones reacted in a very unusual way to an attempted purse-snatching.  When she realized that the boy who tried to steal her purse was young and not a hardened criminal, she decided to teach him a lesson herself.  Instead of calling the cops, she took him home.

When Roger first gets there, Mrs. Jones tells him to wash his face.  He is trying to decide whether or not to bolt.  Being taken home by a strange woman must have been a little scary.

“Then, Roger, you go to that sink and wash your face,” said the woman, whereupon she turned him loose—at last. Roger looked at the door—looked at the woman—looked at the door—and went to the sink.

Roger asks her if she is going to take him to jail.  He doesn’t run away, but he is uncertain of her motivations.  She tells him she isn’t taking him anywhere, and comments that they will have something to eat together.  He tells her he wanted blue suede shoes, and she says all he had to do was ask.

The longer they spend together, the more Roger wants to be trusted. She is a nice lady, and she just wants to help him.  She tells him she was young once, and implies that she made some choices she would rather not have made.  Roger wants her to see the good in him.

He did not trust the woman not to trust him. And he did not want to be mistrusted now.

“Do you need somebody to go to the store,” asked the boy, “maybe to get some milk or something?”

“Don’t believe I do,” said the woman …

Roger’s time with Mrs. Jones does not last long.  When he leaves, you can tell that Mrs. Jones has made an impression on Roger.  He is not going to steal anyone’s purse any time soon.