A video of this demonstration is available at this link.

When you set the two pendulums in motion, the air track cart oscillates about its center of mass.

The center of mass of an object is the weighted mean displacement of all of its mass points from some reference point, usually the origin of the coordinate system in which the object sits. This is the sum of the products of all the mass points with their distances from the origin, divided by the total mass of the object (r_cm = Σm_ir_i/M). It is usually most convenient to set the center of mass of the object at the origin, but wherever we set it, the center of mass of the system that comprises the glider and two pendulums shown in the photograph above, lies at a height somewhere between the glider and the pendulum bobs, on a vertical line descending from the center of the glider.

Whether a system is in motion or at rest, unless an external force acts upon it, its center of mass must either keep moving as it is or remain at rest. For the glider with pendulums hanging from it, then, if you set the pendulums in motion without applying a net horizontal force to the system, its center of mass must stay in the same place.

There are two ways in which you can do this. First, if you raise the two pendulums towards opposite ends of the glider, the center of mass remains on the same line as before (but changes its height). When we release the pendulums, they oscillate in opposite directions while the glider remains fixed.

If you raise both pendulums to one end of the glider, you displace the center of mass towards that end of the system. The glider and pendulums are not moving, so if you release the pendulums by letting the bobs fall, without applying any horizontal force to the system, its center of mass must remain where it was when you released the pendulums. As the pendulums swing towards the other end of the glider, then, the glider moves in the opposite direction. When the pendulums reach the other end of the glider and return to their original position, the glider reverses direction. Thus, the glider and pendulums oscillate in opposite directions, so that the center of mass of the system never moves from where it was when you released the pendulums.