The cylinder is in alluminium, not sensible to the permanent magnet attraction.
If magnet rotates current is produced in the cylinder.
The induced current opposes causes that have generated it.
The cause is the magnet rotation.
For the cylinder, the best opposition is to rotate at same velocity of magnet.
This is basic of induction motor invented by Galileo Ferraris.

The drive shaft rotates because the magnetic field of the electromagnet interacts with that produced by the current that flows in the rotor coil
The current flows in the rotor coil only when the brushes touch the pins that work as commutator.