As we can see in this video, many boxers join each punch with a sharp exhalation, making a sharp hissing sound. It seems to be more common among heavyweights than it is among smaller fighters. Why do boxers do this?
I am an amateur boxer competing so perhaps I can share some insight.
Basically breathing technique is a major aspect one needs to learn. You can very quickly gas out (lose your breath, which also leads to a lack of oxygen for your essential muscles). Therefore, we learn to breathe through our noses when you're not tense - i.e. when you jump-rope and such.
However, the inflow and outflow of air is very different when punching and receiving punches.
First of all, you need to exhale fast when you punch, usually done through your mouth (rather than nose), and through your teeth (you don't want to open your mouth, your jaw can break if you receive a punch). Basically you do it fast when you punch and inhale slowly with your nose afterwards because you don't want a lung-full of air when you get potentially punched in the stomach - I'm sure most people have tried getting the air knocked out of them, and it's terrible and could lose you a boxing fight!
Boxing is about repetition and muscle memory so you do it when doing bag-work and mitt-work as well, to make it second nature.
TL;DR: It's difficult to be punched and you don't want it to happen with a lung full of air.
Exhaling on a strike is common in many martial arts as well as boxing (most forms of Karate and Tae Kwon Do for example).
The purpose of a short sharp exhalation at the end of the technique is to grip the core and "connect" the punch to the body. This helps with both timing and power.
For sparring (and thus boxing as well) this has the benefit of ensuring you continue to breathe and helps mask any symptoms of fatigue (which is important in the mental battle with an opponent)
This exhale can also be done to lessen the shock and pain of an incoming strike. As you become more practised at it a short breath out will tense your core against the blow.
In Tae Kwon Do this is part of the theory of power.
Huffing while punching is sort of a golden rule which they teach in boxing.
The effect of punching without exhaling is much lesser than that of while exhaling. If the boxing classes have very advanced instruments, they show you with the help of a device which calculates the power of your strength. In that it is amazing to find that such a simple thing can cause a great change.
You can try it on a friend. (:p just kidding)
The huff is a quick exhale that brings all the contributing muscles in your body to support that punch you throw. At the same time it prepares your conditioned body to absorb any shock from a counter-punch that could strike your body. This is part of a breathing technique that also ensures a steady supply of oxygen/blood pumped to those strained muscles and to your brain.