If you ignore all the differential equations posted above and just look at the sign of the capacitive reactance you'll notice that it is negative and the inductive reactance is positive and their values are a function of frequency.
In a purely capacitive circuit (one where the resistive component is zero), the current through the capacitor leads the voltage by 90 degrees.
In a purely inductive circuit (one where the resistive component is zero), the current trails the voltage by 90 degrees.
The magnetic field generated by a coil (the windings of a motor), is a function of the instantaneous current.
The windings of a coil in a motor have both reactance and resistance (referred to as impedance). In a motor both the resistance and reactance are positive and hence the magnetic field trails the phase of the applied ac voltage.
If you add a carefully chosen capacitor to the circuit it's reactance is the opposite sign of the inductive hence the net reactance is smaller, and may be positive or negative and the current magnitude and phase are shifted.
So if you add a capacitor to only the starting (usually stator) windings it causes the magnetic fields to be out of phase and applies a torque to the rotor windings and causes the motor to spin.
You can demonstrate this effect with two bar magnets both aligned to lie on same longitudinal axis. Hold one South Pole close to a north pole and the magnets simply pull each other. Next rotate one magnet a few degrees and you'll feel a torque develop.
