The big muscles = legs. If we were doing a broad jump the legs would lead and be the main source of power. But the swing is (mostly) rotational. The legs can assist but are not the main source.
If you go frame by frame here and look at the acceleration (and force) Youll see that the greatest acceleration(and force) is right before contact when his hips are extending.
What's most impressive is that the bat is already going 70 mph and hes still able to accelerate it 15 more mph in a very short amount of time.