Simple: it's a kickass tune with cannons. In 1974 arthur fiedler, conducter of the boston pops orchestra, decided to use it in his july 4th performance. It stuck and only gained in popularity, coz ya know, cannons.

Music sometimes has a tendency to lose the context it's been made in, and it usually doesn't require it's audience to know the context to become popular in the first place. Similarly, Born in the USA by springsteen is usually seen as ultra patriotic. Until you listen closely to the lyrics and realise it really isn't. But if it's a catchy tune and hits the market at the right time, songs sometimes can acquire an entirely new context by themselves.

In what ways do you see the 1812 Overture associated with July 4 or any other American holiday?

I thought the standard for July 4 were John Phillip Sousa pieces, like Stars and Stripes Forever.