1. Not leaving air in your lungs: This is the same reason Judo guys are taught to exhale or yell when they are thrown and as they hit the mat. Having air in your lungs when you fall or are punched causes that sensation of having the wind knocked out of you. Lungs that are full at the moment of impact means you will be left momentarily stunned and gasping for air. When you are punching or throwing any type of strike is when you are the most vulnerable / "open" for a counter. Hence, just like keeping your chin tucked and your guard up when throwing a strike, exhaling is a way of leaving yourself slightly less exposed when attacking: if you do get hit with a counter, you won't have the wind knocked out of you. This is the most important reason you see boxing and kickboxing guys and gals exhaling with every strike they throw. Be careful of how you're doing it though because exhaling by opening your mouth wide open can be even worse than not doing it at all--you'll end up with your jaw on the floor and your lights out. Properly exhaling the air from your belly / abs actually takes a great deal of practice.
2.Tightening your core as you strike: Much of the power from a punch or kick comes from your legs and your core. Tightening those stomach muscles as you throw will add a bit more power.
3.Remembering to breathe: People, especially beginners but even very advanced athletes in a high pressure match, tend to forget to breathe regularly in a fight or competition. Remember we're talking about PROPER breathing DURING a high intensity match where your mind is thinking about a hundred other things at once. If, like me, you've watched thousands of MMA matches, you'll notice every third or fourth match, even for high level, experienced competitors, their corners yelling: BREATHE!!! BREATHE!!! or stuff along the lines of REMEMBER TO BREATHE! This actually happens in other sports too. Any highly demanding cardio activity, especially like boxing and grappling there is a significant anaerobic component as well, which requires that you take in air as often as possible.
4. You must breathe. If you hold your breath, your blood doesn't stay oxygen charged. Bad things (e.g., cramps, significant loss of strength) happen very quickly. It sounds stupid unless you've experienced it in actual competition, but it's quite easy to forget to breathe properly (i.e., taking in air through your nose, quick controlled exhales through your mouth) on a regular basis (i.e., every 10-20 seconds) during a match. We all tend to naturally hold our breath when we're concentrating hard. Finally, remembering to breathe can help some people with their timing. You may not realize it, but the timing and rhythm of your strikes are probably based somewhat on the timing and rhythm of your breathing.
5. Blood pressure: Remember how in high school, your football trainer or PE coach or whoever told you to inhale when lowering and exhale when lifting weights? This is especially true if you're doing low reps and high weight. Holding your breath during intensive exertion raises blood pressure. This is usually not a huge problem if you're young and in good health, but it is good practice and common knowledge. Same applies to a strike if you're putting a lot of strength behind it. It's a single explosive movement that's not so different from a single rep on a bench or squat cage.