This is one of those controversal theories, each person seems to have their own ideas.
The way I see it is thus:
You lose weight as a result of calories burned vs calories taken in. If you exercise at a high intensity (higher then the one suggested for fat burning) your going to burn that many more calories. There are also theories on working out at such a highly intense pace boosting your blood flow for far longer, boosting your heart rate that bit more as your muscles will have been pushed that bit harder, the circulation required to use them will be that bit more. So you burn calories after your workout for that bit longer. This results in a higher calorie burn when your done exercising too.
If your burning more calories running fast then slow, it would follow that your going to lose more weight.
HIIT is something many people do to lose more weight then otherwise. Its used as its highly intense, the body is apparently burning more calories at rest following HIIT then if you were exercising at an average rate for longer. Its also said to reduce appetite too which I cannot comment on (I have not yet managed to get to HIIT standard!) but its another side to exercise: many people will eat more as a result of working out, many assume they burn more then they actually do or disregard the fact that the body becomes used to certain forms of exercise and stops burning as many calories to do so ot trusts machines such as the treadmill which are highly inflated and at best, something to include as a dinner party joke.
The thing I don't get/trust/believe about the slower running = more body fat burned is that if that were so, why do people who do no forms of intense exercise have high body fat? theres a well known thing as skinny-fat where people who do little or no forms of intense exercise (they walk and thats about it) have high body fat around their organs. I can't help thinking that the slow running idea stems from people trying to feel better about taking a more laid back approach to exercise. Yes I agree any exercise is better then none, but theres no reason to boost it up for more then it actually is. I don't buy it.
Like I say, each to their own!