So the truth is that despite the name, they don't really fly, they are just so streamlined and such a perfect shape that when they leap up out of the water their motion, and their long wing shaped fins, carry them forward for a short distance. Of course it is probably the wings that make them appear to be flying as well, because we don't expect a fish to have wings at all.
The reason why they fly is rather a sad one, for they leap upwards to escape predators attacking them from below - and when they are above the water they get spotted by birds looking for their dinner. Their wondrous gift therefore is not, for makes them doubly vulnerable.
By the way "Flying Fish" is a very exciting name, but they are also known as "Flying Cod". Doesn't have quite the same appeal, does it!