This was in part because characters all played off each other very well. Even though the jokes sometimes fell flat, there was enough charisma that you wanted to see what trouble the boys in Bad Company would get into next. The game never took itself too seriously with it's upbeat, jazzy soundtrack and snarky dialog. This is exactly the kind of self-awareness that I love to see in games.
I was really disappointed that they managed to cock up so badly with BC 2 and lose all the personality the first game had going for it.