In an April 5, 2017 article in the New York Times, Gretchen Reynolds reviews new research on the science of slow breathing and how this ancient technique may work to promote relaxation, reduce anxiety, and prevent panic attacks.
The technique of controlled breathing or pranayama (प्राणायाम) is referred to in the Bhagavad Gita, and thus dates back at least to the second century BC.
“Take a deep breath” is the opening to many ancient and modern relaxation and meditation techniques.
Breathing lies and an interesting intersection between the conscious mind and the unconscious mind.
Panic disorder, which involves a dysregulation of rapid breathing associated with an overly strong connection between anxiety and anxious thoughts and hyperventilation, illustrates the complexity of this relationship. Obviously, anxious thoughts are conscious, and these are often part of the trigger mechanism for hyperventilation, once triggered however this rapid breathing remains active until the unconscious brain’s natural homeostatic processes kick in and normal breathing resumes. In the extreme case this happens if a person faints or loses consciousness, but most of the time it takes place as the brain and body reassert a normal balance between sympathetic and parasympathetic activity.
Scientists at Stanford University have identified a small set of brain cells in the brainstem (more specifically in the preBötzinger complex (pBc) of the ventro-medial medulla) that appear to control the balance between rapid breath and slow breathing.
These same neurons play a major role in anxiety.
When these neurons were deactivated in mice, the mice appeared much calmer than normal mice. But otherwise there was no effect. That is until the researcher started to take a look at breathing rhythms.
The mice with the deactivated neurons did not show the normal breathing response (increased rapid breathing) in the face of anxiety provoking stimuli.
It turns out that the deactivated neurons not only controlled rapid breathing but they connected with another part of the brain that plays a key role in sympathetic nervous system activation (the locus coeruleus), and thus affects the entire fight or flight system.
Back to the question of why slow breathing has such a calming effect. The researchers 
speculate that this is because there is a reciprocal relationship between activation of the rapid breathing neurons and activation of the slow breathing neurons, so that if we consciously activate slow breathing, it has the same effect as the scientists genetic deactivation of rapid breathing neurons.
And the deactivation of the rapid breathing cells in turn deactivates the “fight or flight” cells in the locus coeruleus.
It generates calm.
References
Yackle K, Schwarz LA, Kam K, Sorokin JM, Huguenard JR, Feldman JL, Luo L, Krasnow MA. Breathing control center neurons that promote arousal in mice. Science. 2017 Mar 31;355(6332):1411-1415. doi: 10.1126/science.aai7984. Epub 2017 Mar 30. PubMed PMID: 28360327.