Roger probably wants to express his gratitude to Mrs. Jones for her kindness and empathy.
Since he has been jerked around, put into a wrestling hold, dragged up the street, and told that when she is finished with him, Roger will "remember Mrs. Luella Bates Washington Jones," Roger is probably very nervous when he first enters the rooms that she rents. The imposing woman tells Roger to go to the sink and wash his face, instructing him to let the water get warm first. As the water runs, Roger asks Mrs. Jones if she will take him to jail. She teases, "Not with that face; I would not take you nowhere." Then, she grows serious and remarks that she, too, has done things which she would not tell anyone—"[I would not] tell God if He didn’t already know."
After Mrs. Jones rises from the daybed on which she has been sitting, Roger does not want her to suspect that he would steal anything out of her purse, which she has left sitting there. As a result, he moves and sits on the far side of the room where Mrs. Jones can see him as she prepares the meal.
He did not trust the woman not to trust him. And he did not want to be mistrusted now.
Mrs. Jones cooks a meager meal that she shares with Roger. She even cuts the little cake she has bought for herself into two pieces so that Roger may eat some. When they finish their meal, Mrs. Jones rises and hands Roger ten dollars for his "blue suede shoes," admonishing him to behave himself.
Roger is speechless; he wants to say something other than "Thank you, m'am" to Mrs. Jones, but he can barely utter "Thank you" before she shuts the door. Perhaps he would like to tell her that he is grateful that she has not reported him to the police, but has, instead, demonstrated motherly understanding, kindness, empathy, and generosity after inviting him into her home.