I understand that birds think there's another open area ahead of them when they fly into a mirror, but why do they want to smash their head into that other bird?
With the exception of various raptors, birds can't see straight ahead. They wouldn't even be aware of the "other bird" until far too late.
Because they're aggressively warding off other birds from their territory. Essentially they're playing chicken with that other upstart bird in the mirror that won't back down. And they both lose.