Connecting with your breath is a method for being present. When you concentrate on each aspect of the breathing process, you are present; you let go of the past and future and are focused on the moment inside the breath. This is why breathing consciously is its own meditation. But this is just the beginning of why conscious breathing is important.
When you breathe consciously you activate a different part of your brain. Unconscious breathing is controlled by the medulla oblongata in the brain stem, the primitive part of the brain, while conscious breathing comes from the more evolved areas of the brain in the cerebral cortex. So, conscious breathing stimulates the cerebral cortex and the more evolved areas of the brain.
Consciously breathing sends impulses from the cortex to the connecting areas that impact emotions. Activating the cerebral cortex has a relaxing and balancing effect on the emotions. In essence, by consciously breathing, you are controlling which aspects of the mind dominate, causing your consciousness to rise from the primitive/instinctual to the evolved/elevated.
By changing the breathing pattern, you can produce different states of mind. Slowing down the breath has an impact on your emotional state. The cerebral cortex is activated through consciously slowing down the release of breath. Then, the cerebral cortex sends inhibitory impulses to the respiratory center in the mid-brain. These inhibitory impulses from the cortex overflow into the area of the hypothalamus, which is concerned with emotions, and relax this area. This is why slowing down the breath has a soothing effect on your emotional state.