Why does breathing in through your nose and out through your mouth work so well during exercise?

So the reason it's advised to breath in through your nose and out the mouth is that it slows down the rate of breathing. If you breathe solely via your mouth, you run the risk of breathing too fast and reduce the efficiency of oxygen exchange (within the lungs, an exchange of oxygen/carbon dioxide happens at the blood level - if you breath too quickly, there might not be enough time for as much oxygen to bind as there could be. As a consequence, less oxygen is circulating to feed your muscles).
Try right now! Breathe swiftly for about 30 seconds and you'll soon notice yourself getting light headed (don't pass out though please).

TL;DR breathing through the nose slows down ventilation allowing more time for gas exchange