I figured this was a traditional performance, and as such maybe I had it misunderstood. However, from what I have read, this is a fairly recent trend, only becoming popular over the last 30 years. Possibly a copycat of dressage's beautiful piaffe, then? For a horse to perform piaffe correctly, they should appear to be trotting in place. In order to do this, they must build up the muscles over a long period of training to learn how to balance themselves and keep rythm and relaxation. Find more info on piaffe here. It looks a little something like this:
Notice how the horse is lowering down in back, maintaining tempo and impulsion (for the most part- he does lose his hind end just a bit at one point), and is relaxed and quiet in his work? Now let's compare that to this training video of the horse we saw in the first video. Make a note of the way his hind end is popping up. You can also see the chains used to help exaggerate his leg action- the ends are left long to swing up and pop them in the legs to get them to snap up quicker and higher.
Now I'm not an expert here, but that horse doesn't seem to happy and relaxed in his work to me. This horse dancing business, like any other equine sport, surely has it's good eggs and bad. I'm sure there are trainers out there who take the time to develop their horses properly, but the "tie them up and smack 'em till they do it" method seems to be much more prevalent. You can visit this page for some further discussion as well as another video showing multiple horses in action.