A dipole magnet has been at rest on an aluminum disk for a long time. Long means long compared to the magnetic diffusion time of the magnetic field into the disk, which is determined by the resistivity of aluminum and the thickness of the disk. When the disk starts to spin, Faraday's Law requires the expulsion of the magnetic flux of the dipole from the conducting disk. That flux is expelled and the magnet then levitates on a combination of the expelled flux and its own flux.
Another way to think of this is that there must be eddy currents induced in the surface of the moving disk sufficient to exclude any magnetic field from the disk. It is the force of repulsion between those surface eddy currents and the magnet that cause to magnet to levitate.
If the disk stops spinning, the eddy currents dissipate, and the magnet ceases to levitate.
