The disc itself is turned slowly by the force of a spring motor whose speed is controlled by a governor (a spinning fan that maintains a steady rate by the resistance of the air). Some discs are turned by drive holes around the perimeter of the disc, others are turned by teeth along the perimeter and still others, usually smaller diameter discs, are turned by one or two drive holes near the center of the disc.
The discs were cheap to make in comparison to pinned cylinders, and that was the primary reason for the proliferation of disc musical boxes after about 1890.