In literary terms, an external conflict refers to the struggle a character has with another character or with other outside forces such as those in nature, society and so on. Such conflict adds drama to the plot and assists in defining and developing a character.
In Thank You M'am, the chief external conflict is the one between Roger and Mrs Luella Bates Washington Jones. Roger tries to steal her purse, which starts the clash. The conflict is extended when she takes him prisoner by gripping him by his shirt front. Her question about whether he would run if she turned him loose and his reply that he would generate further conflict: she says that she will then not release him.
A further suggestion of conflict between the two is when Mrs Jones declares that she is going to wash his face since he looks so dirty. Roger will evidently have no choice in the matter. This conflict is accentuated when she locks him in a half-nelson and then drags him to her home. There is a clear and direct conflict in this regard: Roger is Mrs Jones' unwilling captive and she his harsh but benign captor.
The conflict between the two is resolved when Roger decides not to attempt an escape and is given food by Mrs Jones. Further resolution is found in her decision to give him money to fulfill his desire for a pair of blue suede shoes, a want which led to the conflict in the first place.
Further external conflict lies in the economic and social conditions both characters face. It is obvious that both are challenged by impoverished conditions, Roger more than Mrs Jones, since she has learned to adjust and cope. A further external conflict for Roger is the fact that he does not seem to have anyone who cares much about him. He has to generally look after himself and he has, in this particular incident, tried to do just that.
This kind of behavior would also bring Roger in conflict with societal norms and values and, obviously, in conflict with the laws and those who uphold them. For Roger, though, things have turned out better for him than he could have expected.