Ignoring the Triac speed control, the motor you describe is a standard "capacitor-start, capacitor-run" type of motor with two stator windings. The capacitor is in series with one stator winding, in which the current leads the current in the stator winding without the capacitor. The phase lead in the one winding vs the lagging phase in the other winding creates a rotating magnetic vector inside the motor, which induces current into the shorted turns in the squirrel-cage rotor and causes it to rotate.
The triac speed control just reduces the overall current in the windings, increasing the slippage at low currents, slowing the motor.