I prefer to call it 'being able to control a horse's hind quarters'.
These are just some of the problems it prevents, corrects or controls:
1) You ask a horse to move his shoulders to the left and instead he throws his a** to the right. This does not happen if a horse has learner to 'yield' his hind quarters to the rider's leg.
2) You ask for a flying lead change and your horse only changes in front. This is because his hind end has been left out in 'left field', so to speak. If you have not taught him to yield his hind quarters to your leg, you have no tools to ask for a 'proper hind-to-front' change much less to ask him to catch up when he drops a hind lead.
3) You ask for a small circle and your horse throws his hind end 'out' and into the outside lead and you cannot make him bring it back in without dropping to a trot and asking for a canter over. If you can control his hind quarters, this just does not happen. [This is what frequently happens to a an improperly trained barrel horse when he goes around a barrel and loses his hind lead -- making it impossible for him to 'drive' off of that barrel to the next one. It will cost him 1 - 2 tenths of a second each time this happens.]
4) You can't figure out how to get a horse to 'look' the right direction and stay parallel to the direction of travel when you try to teach a 'proper half pass'. Instead, you just get a simple 'leg yielding' exercise. You HAVE to have control of the horse's hind quarters in order to get a proper half pass. [If you don't believe this, just watch a high level Grand Prix Dressage test and watch where the rider's outside leg is during a half pass.]
5) In order to do correct flying lead changes on a straight line, the rider MUST have control of the horse's hind quarters. [Again, just watch a Grand Prix rider's leg aids as they do Tempe Changes. The outside leg (for each new lead asked for) is both the aid asking for the change and the stabilizing force that the horse has been taught to yield to if he does not bring that hind leg up and into the new lead. This control of the horse's hind quarters is what keeps a horse perfectly straight while doing these changes on a STRAIGHT line.
Now, what no one seems to understand is that you teach a horse to 'yield' his hind quarters. You teach him to bring them in when you ask so that he knows that he must respect your leg and put his hind quarter where you want them. Then, as his training progresses, you use that control to get him to do exactly what you want with his hind quarters. Obviously, you do not want him to initiate throwing his quarters either in or out. But, if you have not taught him that you can control them, you do not have the tools to get any of the above maneuvers done correctly. You are just hoping for the best and are left helpless to fix the problems -- until you teach him that YOU control his hind quarters.
If anyone want to know the methods I use to get this kind of control of a horse's hind quarters (including bringing them to the inside on a canter departure to teach a horse to drive off of its hind end on a departure on the correct lead rather than hopping up in front to depart), I will try to go through the methods that get this kind of control of the hind quarters.