I'm going to neglect the changes of kinetic and gravitational potential energy changes of the ball for this explanation. What is happening when you punch the ball like in the video is an example of an inelastic collision. When you punch a ball, your arm actually bounces back with some speed, so not all of the energy is lost immediately. Part of the energy has been transferred to the air as sound, and part of it has been transferred to the ball, causing rapid vibrations in the ball. Some of these vibrations "radiate away" as sound for a short period of time after the punch. A small amount can result in permanent deformations in the ball. The rest of the vibrations stay with the ball as "heat." Similar vibrations occur in your hand, propagating up your arm and into the rest of your body, most of which heat it up.
Now your arm has bounced back and still has some energy. To get ready for the next punch (with the same arm), you then decelerate your arm to a halt (or at least to a low speed), doing negative work on your arm in the process. A fraction of this energy may be stored temporarily in your arm as potential energy (as in a spring), to be converted to kinetic energy during your next punch. However, your body is unable to store the rest of this energy in a useful form, so it is lost as heat. This deceleration can also be viewed as an inelastic collision. Moreover, not only can you not recover this energy, since you contract your muscles to supply (at least part of) the force to decelerate your arm, you spend energy in the process, some of which is also lost as heat in your muscles.
Then you spend energy to accelerate your arm, punch the ball, and this repeats.