Though a small-time thief, Roger is still a young boy. Mrs. Luella Bates Washington Jones, an affectionate and sensitive woman, would like to see a positive change in Roger.
Instead of handing Roger over to the police, she takes him to her home. Roger is baffled by her motherly affection. With her every word and gesture, he experiences the warmth of human love, so far unknown to him.
Your question is quite reasonable as to why one would be so generous to a stranger, and especially to someone who has just tried to snatch your purse.
But Mrs. Jones is one of those few people who are unusually caring and big-hearted. She is the type of woman who would do good to even them that intend to harm her. However, Roger was just a boy and she understood he is not a bad person.
Instead of being angry with Roger, she seems worried about him. To her, he is only a misguided lad. Being a thief, he is sure to land up in a jail, sooner or later. She knows the path that he’s taken is self-destructive.
Here’s your answer. By being so good and kind to Roger, Mrs. Jones is taking a chance. Through her love and motherly affection, she wants to appeal to the good side of Roger’s nature with the hope that this could bring about a positive transformation in him.
When she says to him, “You could of asked me,” she means he didn't need to steal. The point that she drives home is Roger doesn’t need to be a thief to make money and fulfill his wishes. There are other ways to do so, much more dignified and self-satisfying ways.