It bolts to the engine's crankshaft in the center and to the transmission's torque converter along the outer edge. Basically it connects the engine to the transmission. It also has a ring gear along the outer edge for the starter motor's gear to mesh with.
It's somewhat common for flex plates to crack around the center. Sometimes the car will still move but it will make a loud banging sound. Sometimes the engine won't spin with the starter so of course it won't run.