Tbh, I used to blow off the breathing thing, but @robert_lee89's tip totally changed my view. Exhaling on the swing really helps you hit harder and stay steady.
Funny how the breath controls timing too, not just power. A boxer times their punch with a sharp exhale, same as a golfer with a putter. It locks your whole body into a rhythm so you don't rush. Ever notice how a baseball player has that little exhale right before the bat connects? Guess it's the same deal for swinging a hammer.