The breath in yoga is referred to as “Prana” or our “life force”. Pranayama means the conscious control of prana, of our life force.
The mind and the body are deeply interconnected. We unnaturally separate them many times during the day - for example, we may plan our next day while doing the dishes, so we are not paying attention to doing the dishes because we find something “more important” to think about. We may walk on the street and not observe the surroundings, instead, think about where we need to arrive. Etc. Like this, we dissociate our mind from the present moment, sometimes even for days in a row.
The best link between our mind and our body is our breath. Our breath usually follows the flow of our thoughts - for example, someone who is very attentive and calm will usually have a deep abdominal breath and a regular pattern of breathing; by contrast, someone who is angry, worrying or being anxious will breathe from the chest and will have a shallow, irregular pattern of breathing.
The best way to switch between states is to become aware of our breath. Our breath is the easiest link we have with the present moment because it is continuous, always there for us to observe. When we observe our breath, we are disrupting our thought patterns and bringing our mind and body together in the present moment. By taking control of our breath, we are taking charge over our mind and over how we respond in different life circumstances. We are also training our capability to become attentive whenever we choose to.
This is why training in becoming aware of our breath and training our ability to focus on using our breath is so important and emphasized in both yoga and meditation.