At the start of each screen you'll find various buildings.  Blow fire
over them, and then hit them.  This will make them fall down a lot
faster.  When you burn or knock down a house you get an egg.  To beat
the other monsters in a fight for an egg, don't jump around and breath
fire, just hold your ground and punch forward.  After three or four
falls, your foe will stay down.  Keep eating, even if it dosen't show
any increase on screen, your health will skyrocket.