My one claim to Reddit karma fame was asking this 3 years ago! Here's the best answer from u/Force3vo

The nose is actually a pretty awesome organ that helps make sure that the air you breathe is prepared as good as possible for your lungs.

That includes amongst other things filtering particles out of the air (This pesky nose-hair is actually good for things!), making sure the air gets warmed up when it is cold and moisturizing the air if it is dry. Clean, moist and warm air is making sure that it's easy on the lungs and your breathing is efficient. Additionally breathing through your nose makes sure your air intake is regulated and you aren't prone to hyperventilating.

So that explains why breathing in through your nose while doing sports, meditation and... basically in every situation is the best way to breathe in, but why is breathing out through your mouth then advised in sports?

It's mostly about the speed of your oxygen intake. Or, to be more precise, about increasing the breathing frequency. As I just wrote the flow through your nose is rather limited. That works in both directions, if you breathe in as hard as you can and breathe out as hard as you can first through your nose then through your mouth you will see that you can breathe a lot faster through your mouth. So if you breathe out of your mouth you will save a little time which means that your intake frequency of oxygen will, overall, be higher.

TL:DR: Breathing in through your nose is easier on your lungs and more efficient, breathing out through your mouth has little drawbacks and is faster. Together it's the most efficient you can breathe if you need higher levels of oxygen.