I was thinking velcro all along - until that last one. Pretty cool!
If you look at around 57 seconds, you can see the purple gown fall down over top of her teal coloured one. Apparently they work by thin panels of fabric falling down from the top of the outfit, covering up the last dress (which is also why the dresses get progressively longer for the most part).They make it look easy, but I bet those clothes would be nearly impossible to make. Oh and I remember someone dug up a patent application showing how the quick change clothes work...still, they do an awesome job!