When you breath properly during a yoga session you are providing your body with sufficient oxygen so that the body and its cells function correctly and efficiently. Without the right amount of oxygen, the cells in your body won't metabolize properly, especially food. This means that nutrients that include vitamins and minerals will just be wasted. So, proper breathing is going to allow your body to metabolize food properly and rid itself of noxious by-products of that metabolism. It also nourishes your muscles and organs.
Another benefit of the breathing in yoga is that it produces enough oxygen for the brain to function better. The brain requires a lot more oxygen the other organs in the body and when there isn't enough oxygen going to the brain it can cause you to be sluggish, fatigued, have lack of concentration and so on.
Breathing exercises can help to keep both sides of the brain in balance with each other. The brain helps to control the opposite sides of our bodies as well. The two halves of the brain also control the different functions and aspects of our lives as well. For example, the right brain is the calm side, intuitive side and is more inner directed and deals with emotions, spacial as well as nonverbal activities. Where as the left side of the brain is more aggressive, is more logical, is outer directed, rational, is more objective and deals with the sequential reasoning, mathematical as well as verbal activities. So, when you breath properly it will help these two sides of your brain to work together much better.
Breathing exercises will also help your mental and emotional health. In yoga, breathing exercises are very important and breathing is thought to have a link between your physical and mental aspects of your life. It cleanses and strengthens your body and mind.
Proper breathing can also sooth your nervous system, calms you mind and can improve your concentration and help you focus better so you can pay closer attention to details. It can also lower the effects of stress. It can also help to combat depression, grief and sadness.