Most probably, during a yoga class, you've got instructed to 'just breath', ' breath deeply', 'breath it all out' etc. I remember the first times I heard this and got more stressed than I was before (stressing myself that I was "wasting my time breathing instead of doing actual exercises to become stronger and fitter…."). Little did I know.
You find increasingly more research about how the breath influences our brain. Why do we need to research the breath since we do it automatically, you would think. Well, if you knew how many times you stop breathing per day because of stress for example, and how taking in a few deep breaths can alter your focus and energy level, you wouldn't ask that. You would ask, 'how?'.
One recent study (Jose J Herrero et al., 2018) shows that exercises involving controlled breathing and merely focusing on one's breathing, can alter the connectivity between parts of the brain, allowing deeper access and synchrony between them. Controlled breathing exercises? Simply put: it involves taking full breath cycles, enabling a full oxygen exchange for a longer amount of time. Ultimately, you may benefit from greater (emotional) control, focus and calmness.
It is known that the breath is controlled by the sympathetic nervous system (you know, the one which responds to stress with the flight-or-fight mode). When you do controlled breathing, you counter this mode with your parasympathetic nervous system, the ones that calms you down and counteracts stress. We evoke the relaxation response, and thereby lower stress levels, and like a study (Manoj K. Bhasin et al., 2013) concludes "enhances expression of genes associated with energy metabolism, mitochondrial function, insulin secretion and telomere maintenance, and reduced expression of genes linked to inflammatory response and stress-related pathways."
A yoga teacher may simplify it by saying, it has positive effects on your immune system, stress, energy levels and focus.
So, how? Pranayama (controlled breathing, or breathwork), meditation and repetitive prayer as part of the yogic learning, has been advised for millennia - it is a powerful tool we are re-discovering with the increase of interest in yoga. There are many methods you can learn, the easiest to start with is during meditation: pay attention to every breath, doing full breathing cycles, starting and ending in the belly. Those can be an effective re-energizing method after sitting for too long. Or when you need to regain focus to take a decision...
So, when someone gives you the advice to take a deep breath the next time, you might do it with a smile on your face, knowing that it actually is a good advice.
Jose L. Herrero et al. 2018 [https://www.physiology.org/doi/abs/10.1152/jn.00551.2017]
Manoj K. Bhasin et al., 2013 [https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0062817]