The Redskins are a very old team (roughly the equivalent of being one of the NHL's Original Six). The Redskins became the de facto professional team of the South, partly because of location (all the other teams were more to the north on the east coast, in the industrial midwest, or a bit later in California) and partly because the team's owner was a southerner who aggressively marketed them to southern fans. He owned a bunch of local TV and radio stations which showed the Redskins well before NFL games were inescapable on TV, sort of a proto-RSN, and had the team play exhibition games in southern states in the offseason.
Plus the team was generally relevant in the 70's and 80's as the NFL exploded, while the relative newbie Atlanta Hawks were terrible, and Carolina didn't get a team until the 90's, so that helps.
So while the DC area has a deserved reputation for being a lousy sports market, which lost baseball twice and is thought to have too many itinerant residents to support any real loyalty to the local teams, the Redskins are an insanely big deal here.
edit: A
story on the Skins southern legacy. Coincidentally (ha ha ha) they were also the last team to racially integrate.
