Pilates is well known for its focus on breathing. It's in fact one of the key principles as highlighted by Joseph Pilates, and movements are carried out with focus on what breath does.
The reason for this - and this is pure genius, as Pilates was well ahead of his time and research has only just caught up with him - is that well executed breathing is the absolute key input into the stability of the pelvis. Why would that be important?
The pelvis in a human body, for us bipeds (i.e. creatures walking on two legs), is our centre of gravity. Stability of that is ensured by postural, or core stability, muscles. Our limbs attach to our torso and are moved by large mobiliser muscles. These muscles don't work in isolation - the brain assesses and links their function to the underlying pelvis stability. It is all connected. If mobiliser muscles have to work off an unstable base, they will have a lot more work to do to achieve the same result. In fact our brain might even perceive that the range of movement off an unstable base is not safe and it will limit the range of motion by making joints stiffer.
Good breathing function achieves the right amount of tension which pulls the underlying stabiliser tissues to the right degree of tautness when required, with symmetry when needed. It's a beautifully orchestrated performance, and failure of one of its components will lead to a lack of full stability elsewhere. So get your breathing right, and the body will have to work less hard to achieve its movement and performance goals.