What kind of DC motor is this and what does the circuit do?
That looks like a DC motor of the type used in tape cassette players.

Why does it need that circuit and what does it do?

For music reproduction with accurate pitch constant motor speed is required. Achieving this in battery powered equipment requires addition of a voltage regulator or speed controller as DC motors' speed varies (nearly linearly) with voltage. The potentiometer in the motor allows the speed to be factory set.
I modified one such motor on a good tape deck, adding an external pot and switch to allow some pitch adjustment so that I didn't have to keep retuning my guitar between tracks / albums.

The TDA1151 seems to be one simple device intended for such applications.