This kick is my evil nemesis. I was doing it poorly a year ago. One of our instructors was kind enough to pull me aside and work with me one-on-one to improve it. We were succeeding! Then I really threw myself into it with great enthusiasm! As I landed the kicking leg, I profoundly hyper-extended the kicking ankle inward. The soft-tissue damage was legendary. They x-rayed my ankle twice because they couldn't believe the first x-ray: no broken bones. I was out for months anyway with a bad case of torn-everything and pulled-everything-else. That's when I wrote the wiki, because all I could do was sit in front of TVs or computers. A year later, that ankle still aches.
As my best friend (also a TKDan) is fond of saying, "there's a difference between 'simple' and 'easy.' Even climbing Mt. Everest is simple. You put one foot in front of the other until you get up the mountain. But that doesn't make it easy."
I come from very 'earth-bound' people. I have very little "up" in me (and as I get older, I have less). I leave that to the young, fit, & flexible. God love 'em!
Key to this kick are the timing of the first leg thrown relative to the push off with the standing leg (this is key to many airborne kicks), the amount of spin generated at takeoff, and what the arms do in the air.
Only the arms can help once you're in the air, pulling in to increase spin (conservation of momentum).
It is really a knack, and just requires reps without too much effort.
A good practice exercise is to make that first leg a lazy jumping side kick, then follow through the spin to land on that kicking foot, attempting to land in balance for a back hook kick.
Try this (Example uses left leg kick since that is what the video above shows.)
Picture yourself in the center of a clock dial.
Left leg back / right leg forward and right leg will do the kick.
Without jumoing step with you left foot to 2:00 -3:00 - See video above (You have now turned 240 -270 degrees - each number increment on the clock representing 30 degrees)
Make sure your right arm / shoulder has moved past the 2: 00 mark - One way to do this is place that arm behind your back as you rotate.
Now pivot and kick.
If this flows with the upper body rotation being of paramount importance instead of stepping with the left, it is lifted hard toward 2o as the precurser to the jump / launch with the right foot.
(Easy to show - Harder to expain in writing)