In the western world, the understanding of breathing is that it supplies Oxygen to the body and brain. Yoga maintance that breath carries a very special agent, very vital to the health, this agent is called Prana in Yogic term. Ki or Chi in Japan and China. Without prana, universe will not exist. Breath is a communication line between the body and mind. Yoga has many pranayama (sci of breathing)  techniques for achieving various results for mental and physical health. remember people use to harness the wind energy to sail the ship?  Air/wind (prana) has a huge energy and yoga has learnt to harness it for the benefit of the people.