Breathing is an important part of Pilates for many reasons.
Personally I use it to a help in connecting my muscles, on exertion to complete a move that I may need more strength for, or indeed to actually find that strength. It aids my concentration and focuses my attention on the task at hand namely to move my body and with it comes a meditative state of mind. Hence that well-known phrase "find your mind body connection".
In Return To Life Joseph Pilates often refers to the art of breathing. One of his main points :
"Bodily House Cleaning With Blood Circulation
True heart control follows correct breathing which simultaneously reduces heart strain, purifies the blood, and develops the lungs. To breathe correctly you must completely exhale and inhale, always trying very hard to squeeze every atom of impure air from your lungs in much the same manner that you would wring every drop of water from a wet cloth.........."
Every exercise he lists in his book comes with breathing instructions.
The art of breathing within the Pilates System is so important that the second book on Pilates, written by some of his students , with the aid of his protégé Romana Kryzanowska :The Pilates Method of Physical and Mental Conditioning
lists breathing as one of the 6 principles of Pilates.
"Soon the entire body is charged with fresh oxygen from toes to fingertips, just as the head of steam in a boiler rushes to every radiator in the house."
In my own classes I often end with a breathing exercise I first learnt from a first generation teacher Lolita San Miguel
Breathe in for 5 and out for 5
In for 5 out for 6,