There are a number of different tricks used in this kind of act. If you freeze frame a lot on the scene where the woman changes from a greenish/turquoise dress to the purple one, (she's in a hoop with purple strands of material), you will see that the shoulder straps change colour first (the man has his right hand behind her back and is releasing a catch or fixing stitch).
Then the skirt falls and changes colour (bottom of picture). The purple skirt is pulled up and fixed over the shoulders like a tabbard, it has the turquoise material on it that we see as the short skirted costume. When the fixing is released the pulled up skirt falls down from the shoulders, the purple material (that was previously on the inside) is now on the outside. Irt's not easy to explain but freezing certain areas will give you some clues.
Notice that many costumes go from a short skirt to a long skirt, then the next change is in a bag or behind a screen etc so that the whole costume is discarded to reveal the next one being worn underneath.