What is the meaning of "run someone out of town"?

The expression does have (U.S.) "Wild West" connotations, and it basically means to "eject" someone from town.

This usually occurred when someone was carried out of town "on a rail" by a mob, often after being "tarred and feathered" (i.e. covered by the same). This process was so forceful, and attended by so many people, that only the most intrepid or foolhardy would ever set foot in town again.

The context appears to be a modern one, in which a woman probably got together a "gang" of citizens (in e.g., a "recall" motion), or possibly a "posse" of lawmen, (in e.g. a corruption case), that had the effect of not only ejecting the mayor from his office, but essentially forcing him to leave town altogether.

Most obvious meaning is that she caused the mayor to have to leave said town by or as if by force; either physical, or perhaps by sullying his reputation to the extent that he feels he has to leave, as he's been given such a bad name.

In modern context

To harass someone so much that they are forced to leave the area to escape harrassment.

This is generally a response to unacceptable behavior.

It is also used in context of ending or preemting challenges to political or social standing.