Short answer
Our breathing strongly influences how we think and feel moment by moment. This is because the state of our nervous system is fundamentally linked to our breathing rhythm. Depending on whether we are breathing in or out, the state of our nervous system changes. Different brain areas become more active or less, creating brief shifts in our state of mind with every breath. This link is strongest when breathing through your nose. Intense breathing practices can amplify these effects, leading to profound changes in consciousness and perception.
Longer answer
The brain’s conductor
The activity in many brain areas follows the breathing rhythm – going up and down with our breath. Thus, breathing can be seen as a conductor pacing an orchestra, with different musicians being different regions of our brain. Importantly, in an orchestra, usually not all instruments are played at the same time, though they may all keep the same rhythm. Similarly, many brain areas follow the breathing rhythm, but some become more active when we breathe in and others when we breathe out. This has consequences for how we think. When we breathe in, we are usually better at noticing things, such as facial expressions. On the other hand, when we breathe out, we tend to be better at initiating movements. In this sense, every breath is like a small process-then-act cycle.
It is important to note that this orchestration of the brain is tied to how we breathe. The previously described conductor function applies almost solely to nasal breathing (through the nose). There is almost no connection between our breathing rhythm and brain activity when we breathe orally (through the mouth). This is probably one of the reasons why most meditative slow-breathing practices use nasal breathing.

Breathing can be thought of as the conductor of our brain regions, tuning some regions up and others down. Music director & conductor by Chalumeau73 (Wikimedia Commons), licensed under CC BY-SA 3.0
Activated versus relaxed states
In addition to brain activity, nasal breathing also changes the balance in our so-called autonomic nervous system (ANS). The ANS is a network of nerve cells controlling all the bodily functions that happen without us thinking about them consciously, such as our heartbeat, body temperature, and sweating. The ANS has two major branches, the fight-or-flight (sympathetic) and the rest-and-digest (parasympathetic) branch. Breathing in increases sympathetic activity, while breathing out intensifies parasympathetic activity. This means that we feel more activated on strong inhales and more relaxed on strong exhales. That is why breathing techniques with stretched exhales tend to promote relaxation.

Image by Diana Light via Pexels
Intense breathwork
Finally, intense deep-and-fast breathing (so-called high-ventilation breathwork) can fundamentally change the way we see ourselves and the world around us. By breathing out CO2 quicker than usual, we can increase the pH of our blood (making it a bit less acidic). At the same time, the drop in CO2 causes blood vessels in the brain to constrict, reducing blood flow to the brain. Together, these changes alter the activity of brain cells and brain networks, deeply shifting the brain’s processing mode. When we breathe intensely in this way for more than 10 minutes, we may experience strong, unfiltered emotions, memories and insights, as well as big changes in our sensory experience, such seeing or hearing things that are not there. As such, intense breathwork can induce experiences that share some features with psychedelic states (read more about the effects of psychedelics here). This more extreme example shows that breathing and the brain are deeply intertwined and can shape each other at any moment!