Why does a spinning skater speed up when she pulls her arms closer to her body? It's the same phenomenon that causes the carousel in the video to rotate faster when the students move towards the center. And that phenomenon is in fact one of the fundamental principles of physics, known around the planet as "conservation of momentum." Here, in particular, we have a beautiful demonstration of conservation of angular momentum.
Angular momentum is the momentum contained in an object's rotation. Linear momentum depends on the linear velocity of an object and its mass. However, angular momentum depends not only on the (angular) velocity and the mass of the rotating object, but also on the distribution of that mass.
Let's say you have two wheels of equal mass. If one of the wheels (let's call it wheel A) has most of the mass distributed towards its circumference, it will have a greater rotational inertia than wheel B, which has more of its mass near the center. A will be harder to get rotating than B, but once rotating it will be harder to stop. (You can try this at home with bicycle tires and globs of clay. Put equal masses of clay on each tire. With A put it all on the outside of the tire, but on B put it on the spokes near the center. Notice how much more difficult it is to get tire A spinning.) If both wheels are rotating at equal angular velocities, then A has greater angular momentum, due to its larger rotational inertia.
The principle of conservation of momentum states: In the absence of external forces or torques acting on an object or system of objects, the total momentum of a system will remain constant. To get the carousel spinning, the students exert external torques on it, thus speeding it up. They are also adding mass to the system when they jump on, but once everyone is on the carousel, they become part of the system. When they move to the center of the carousel they do not change the mass of the system, and they also can't exert any external forces or torques on the system. However, they do redistribute the mass. Because they move towards the center they decrease the rotational inertia of the system. In order for angular momentum to be conserved the wheel must therefore speed up -- and, as we can see, it does.
Adam Weiner is the author of Don't Try This at Home! The Physics of Hollywood Movies.