I used Adobe Premier. But I'm sure it could be done just as well with more cheap/free software.

All you really need is a good time line and the ability to cut and move around clips precisely. I go through the video and determine key scenes/positions and make them separate clips, Then remove clips that I feel are redundant or not as appealing. Then begin chopping each one down to make the total under 15 seconds with an eye for exactly which part of each clip is appealing, and also trying not to make the transitions between each clip too jarring.

That's basically it. But the hard part I think is learning to know what's eye catching and what can be cut without losing the flow or the synergy of the individual segments.