Roger places himself in direct physical conflict with Mrs. Luella Bates Washington Jones when he tries to mug her. Unfortunately for him, Mrs. Jones is not a woman who is easily taken advantage of, and she uses this conflict as an opportunity to offer Roger some guidance in how he should treat people.
After dragging Roger to her house (which, it should be noted, he ultimately allows her to do), Mrs. Jones doesn't foster the conflict. Instead, she offers him a place to clean up, and after learning that he doesn't have anyone at home, she feeds the same boy who tried to rob her, even offering to give him money to run to the store if he needs something else for their dinner.
The larger conflict and the one which propelled the conflict between Roger and Mrs. Jones initially is the conflict between Roger and his society. He lives on the fringes, largely invisible, unloved, and without respect. He isn't yet hardened by this environment, as evidenced by the respect which he offers Mrs. Jones. Roger doesn't have anyone at home to remind him to wash his face, let alone anyone who can help him earn the money he needs to purchase the shoes he wants. Without guidance, Roger tries to create his own fortune by stealing from someone else. Mrs. Jones shows Roger another way to deal with this conflict. She tells him that he could have just asked her for the money, which Roger undoubtedly never considered. She's also shown him the very real humanity behind his floundering efforts toward crime, and hopefully this encounter changes Roger into a man who deals with the conflicts of his society in more productive ways.