When actually playing the game, you assume at first glance he's just jumping up and bumping them with his head. The publicity drawings clearly show his fist, though. So it's just that in the game sprites, his fists aren't as prominent as his head, so it just looks like he's bumping his head.
(This video suggests he uses his head, as he ends up bloody (a window ledge us used for floating blocks)).
(I don't know why they added that soundtrack over it. It used to use regular game sounds and music).