Traditionally when we think of our core, we think of our abs... but the core is so much more than that. It's a complex group of muscles that work together to keep you physically stable.
No matter what muscle we are targeting at the barre, our focus is on posture and form... enter the core. It's the powerhouse that allows you to keep your body quiet so we can isolate the glutes, the quads, etc.
Engaging your core in this manner helps build and maintain good posture, which is essential for eliminating low back pain. It's also why the ab set is brief at the end of class. You've been working that core the whole time and didn't even know it.