As the motor rotates, there will be positions where the commutator connects and disconnects the motor coils. These will produce sudden changes in motor current (much too fast to measure on a meter) which cause interference to radiate from the motor wires. This would cause crackling on nearby AM radio and could pick up on a nearby audio amplifier.
The capacitor acts like a small reservoir which helps to even out the spikes in current and reduce the interference. It plays no part in the operation of the motor itself, it is there just to kill the interference it creates.
Brian.