Martial arts are often associated with yelling and screaming, from the kiai of karate to grunts to Bruce Lee's catlike yowls. In the FMA, however, we don't use the kiai much. I've heard masters flat out joke that "We're not karate!" What, if any, is the function and purpose of the kiai?
On a basic level, it is a timing device. It focuses our mind to what we are doing, unifying our movement to our intention. While there are various sounds preferred in different schools, a general idea is a tight or compressed sound that opens and releases energy at the moment of impact. While one might be told that it is just a word, and use another particular sound, kiai is actually onomatopoetic, meaning a word that actually suggests the sound to which it refers (like murmur or buzz). Thus one could say keeeee (smack!) ahhhhhh in delivering a hit.
Using the breath with a hit also releases energy. The unity of mind/body maximizes the impact to the target, but it also can create internal stress in the body, which over time can have negative effect on health. By releasing the breath, we provide a safety valve for the energy of that "equal and opposite reaction" to blow off harmlessly, as opposed to shocking our own internal organs.
Think what happens if you exhale too soon. You have no energy. Hold your breath too long, you restrain yourself.
The sound itself conveys power. There are stories of old masters who could drop birds from the air by stunning them with a powerful yell. I myself once stopped the charge of a huge bull mastiff with a spontaneous yell from the pit of my stomach. It came from nowhere, a combination of terror, inspiration and adrenaline. The power of that moment stays with me, and it's why I'm here to write this now.
Here is a link to an article I found on Stephen LaBounty's website which is more in-depth on developing and using this. There are other interesting looking links on there, such as "The Knife in Combat."