The transverse abdominal muscle is the deepest layer of muscle within your abdomen and is located under your inner obliques. It is responsible for helping you maintain pelvic and thoracic stability while you're working your twisted body.
So, while you're on the mat doing double leg stretch your transversus is keeping you from scooting off of it and out the door. If you're on the reformer pumping out the hundred, the transversus keeps the carriage stable. And if you're working the pole transversus keeps your legs up as you spin into the chair.
There, you've done it! Now remember to breathe and enjoy the burn!