This paper describes the effect in some detail. Stable levitation is caused by a combination of the Meissner effect, and of flux pinning in Type-II superconductors.
The Miessner effect is the property of superconductors which prevents magnetic flux from penetrating the superconducting material (beyond the penetration depth). It occurs because the applied magnetic field induces "shielding" currents in the surface of the superconductor, which generates an opposing magnetic field that acts to exactly cancel the applied field. It is as though you've taken two conventional magnets, with the two north poles pointing each other, and suspended one above the other.
This alone is not enough to produce the stable levitation that is shown in the video; that requires the flux pinning property of Type-II superconductors. Flux pinning occurs when a magnetic field of relatively low strength is applied. The flux does partly penetrate the superconductor, but is concentrated around imperfections in the material. The superconductor is held in position rigidly in the magnetic field, and experiences a kind of friction when nudged because flux vortices are forced to move from one pinning site to another.
The design of the track allows the superconductor to move freely in one direction:
Along the length of the track there is no variance in the field, which allows the superconductor to move back and fourth with no energy loss. Perpendicular to the length of the track, the bar magnet's poles are aligned anti-parallel to each other, (S-N-S). This alignment produces a considerably strong gradient [along the width of the track]. The variance of magnetic field strength from one side of the track to the other is so great and the pinning so strong in this superconductor that there is not only drag but also a restoring force. If the superconductor is given a small push in attempt to force it from the track, it will oscillate slightly and quickly return to its original position.
If there is enough pinning and if the magnetic track is strong enough, then the levitating "train" should remain locked in place even if the track is rotated sideways or is upside down. Note that if the applied magnetic field is extremely strong, this can cause the superconductor to loose its superconductivity.