Breathing as I mentioned is super important. I first saw breathing implemented in martial arts when I came across Systema which is a Russian hand to hand combat system. Years after that saw the Gracie BJJ practitioners use breathing as a part of their training as well. There are many breathing exercises from various forms of training that can be used. BJJ uses some of the ancient yoga pranayama exercises.
Our minds and our minds are connected and you will notice how your breathing pattern can change as a response to environmental triggers which can lead to emotions such as fear and stress. Our bodies make hormones such as cortisol (stress hormone), growth hormone, testosterone, VIP, etc. Research has shown that there is some correlation between breathing patterns and hormonal changes.
In video below you will see a Gracie BJJ practitioners demonstrate some breathing exercises and talk about the benefits.