You could write it off as blind chance if, say, it worked for three or four elections ... but the rule has incredibly held true for every damn election since 1940, except one (and we'll get to it in a minute). So this is slowly entering gypsy curse territory.
Because why the hell would this possibly work? Maybe you could say that the incumbent is re-elected when things are going well for the country, and when things are going well, the crowd will be more jazzed to root for football, and the positive crowd makes the team play better. But why would it only apply to that one game? Keep in mind that it has nothing to do with how good the team is overall -- the 1976 Redskins only lost two games at home all year, but by God, one of those two losses was right before election day, and therefore the Republican incumbent lost and Democrats took back the presidency. What the hell?
As for the one exception, it was in 2004, when the Green Bay Packers beat the Redskins but George W. Bush held on to the presidency. The fact that this was the one exception actually makes it weirder, because as some of you vividly remember, Bush was president but had actually lost the popular vote in 2000 (winning only due to the Supreme Court craziness over Florida's recount). As the guy credited with discovering the theory, Steve Hirdt, points out, if you make the rule refer not to the party in power, but to the party that won the popular vote in the previous election, it suddenly has a perfect 18-for-18 record predating World War II.