Why do we blow when we're out of breath?
Trying to rephrase the question

If I understand the question (and what you missunderstand) correctly, I think one could rephrase the post as

When one spends some time under water and come back to the surface, one usually breath out and then breath in. This is surprising to me because I would expect the person to be out of air and therefore wouldn't need to breath out before breathing in.

Let me know if I failed to understand your question and misunderstandings

Answer

We don't incorporate the air we breath in our lungs. We only use the molecules of oxygen $O_2$ from the air (which takes ~20% of the total volume). This molecule goes in the blood, reach the cells and react with sugar to produce energy. The waster of this reaction is another molecule $CO_2$ that needs to be exhaled through the lungs.

So, really the volume and of air that we have in our lungs doesn't vary much (It actually increase a little bit as we had some carbons). Only the composition of the air that we have in our lungs changes. After some time, we need to renew this air and for this purpose we first need to breath out the air that is contain a high density of $CO_2$ before breathing in.

This is why when sticking the head out of the water many people first breath out and then breath in. Note, however that one can breath out all its air underwater and then only has to breath in when back at the surface.

EDIT

Are you asking why do we contract our lips during sport?

If yes, I think it very much depend from one person to another. There is probably no physiological advantage/disadvantage at doing so and it probably is just a consequence of expressing pain on our face.