Breathing is a majorly significant aspect of martial arts: both external and internal.
But it represents something more than just air intake and focus... it is a metaphor for training as a whole.
Expansion: The taking in of air. This is the time where we devote ourselves to countless hours of taking in diverse techniques, information, trials, and experiments. It is the state of mind where we devour concepts and open our beings to new insights and styles. It is the feeding of the martial body and mind, just as breath invigorates our blood and tissue.
Contraction: The exhalation. This is the moment where we clear our minds and focus our attention on all the things we've taken in, discard what is not to our liking, and fine-tune the lessons we have learned during our cycle of learning/expansion. It is the realignment, where knowledge is turned into understanding. Just as air is cleansed and filtered through our lungs, then given as oxygen molecules to fuel our system.
In yogic tradition, all things are accomplished in cycles of breath, namely the universe itself. It is considered to be in a constant state of increase and decrease, comparable to breathing. All the worlds and stars are created in this process, just as when one meditates, worlds of thought and energy emerge in our minds.
To the martial artist it is all a part of the spiral of learning.