What are the DC motor wires in the picture for

It will be a Brushless DC motor, presumably with Hall sensors (the twisted cable).

Hopefully that should point you in the right direction in looking for a drive solution - there are many brushless DC drivers out there - sometimes called ESC's. But essentially they comprise of three half-bridges, one driving each phase. The Hall sensors feed back the rotor position to allow for correct timing when commutating the motor (switching phases digitally) without needing to resort to BEMF to determine rotor position which tends to be more tricky to do.

EDIT:

As Nick rightly points out I forgot about power when I was counting wires for the sensors. The twisted cables are probably some for of speed feedback. In this case you would need a driver which is designed to run BLDC motors which performs the commutation based on EMF from the motors.

The relay is likely connected to an auxiliary motor winding that has a different number of turns, a different wire size and resistance compared to the main winding. That would act to start the motor similarly to capacitor starting. There are several possibilities for the motor design details. It is not possible to completely and accurately explain the motor design without close examination of the construction. Photographs might help, but that might still not be enough to determine exactly how the motor works.

Edit Re Photos

Note that the relay has relatively few turns of heavy wire. I believe it is a current relay, wired in series with the main winding. When the motor is first turned on, a high current flows in the main winding through the relay coil causing the relay to connect the auxiliary winding. Since the aux winding has a different inductance and resistance, its current is phase shifted compared to the main current similarly to the phase shift that a capacitor would cause. That defines the direction of rotation and provides starting torque. When the motor approaches full speed, the current drops and the relay disconnects the aux winding.