If space is a vacuum and there is no atmosphere, what does a rocket engine “push” against to propel the vehicle? It's always been a head scratcher for me.
It's not pushing against anything. It's expelling mass (rocket exhaust) in one direction and Newton's third law (conservation of linear momentum) says that the rocket has to move in the opposite direction. Exactly the same thing is responsible for the "recoil" or "kick" when you fire a rifle. It's not because the bullet is "pushing" on anything, it's because the bullet left the rifle barrel at high speed and the rifle moves in the opposite direction with the same momentum as the bullet (which, because the rifle is much heavier than the bullet, is a lot slower in velocity terms).