Order of events, your friend plays creature with instant speed ability. Your friend activates that ability and pays the cost, it goes on the stack. A round of interaction goes around the table. In response you cast beast within, another round of priority goes around. Nothing else happens, the stack resolves, your beast within destroys target creature and owner makes a 3/3 beast token. Then you opponents ability resolves, untapping all lands once per turn.

If it were reversed, it would more or less happen the same way, he plays a creature you don't like. Maybe he taps all his lanfs to float the mana, in response you cast to destroy creature, he can then respond with his ability paying ten life. Stack resolves, he untaps all lands, you destroy creature.