The kiai, kihap, or "shout" serves many different purposes. It can help provide focus by association (you shout when you strike in practice, so shouting in combat helps you land that prototype strike). It can help provide power (I don't know the mechanism exactly, but shouting or grunting often helps people exert more effort, something to do with activating your abdominal muscles). It can startle or intimidate your enemy before, during, or even after the strike. And I've had a few people say that forcefully exhaling like that helps you expel carbon dioxide from your body in preparation to breathe deeply and reduces your chance of getting the wind knocked out of you.
Personally, I usually do my kiai either right before the strike to startle an opponent (it provokes a startle reaction in a surprising number of people, even the trained ones, that can disrupt their guard and leave them flat-footed, or swinging early before you're in range) or during to increase the power and disrupt my opponent's concentration on their defense.
After actually watching the video (sorry...), I've seen examples of all three. The kicker gives an abbreviated kihap right before kicking, often used to firm up the stance/chamber before kicking or to startle the opponent. They also sometimes do the kihap during the kick or at the time of impact to increase the power of the kick. And yes, a few times, he says the kihap after the kick, which I believe is him trying to indicate that it's part of the impact, but due to the slow demonstration speed of the kick, it's delayed, much like how in a demonstration an instructor might place their strike, then say "strike" before moving to the next move in the sequence.