Why do you connect a capacitor to an AC motor?

To induce a phase shift between the rotor and stator (stationary winding).

AC motors are not good at starting up, they need 'help' to get started/they need two magnetic fields to push against each other to generate torque. This capacitor is called a 'starting' capacitor and provides an extra 'boost' to get the motor turning by increasing the phase angle between the rotor and stator winding. Once the motor is at it's proper operating speed, the capacitor must be disconnected or it will burn up.

There are also motor designs that use a run capacitor. This capacitor usually has a smaller capacitance than a start capacitor (so it provides a smaller phase shift), but is designed for continuous operation. These motors don't provide as much starting torque as a similar motor with a start capacitor. Other motors will have both a start and run capacitor. The start capacitor provides significant phase shift between the rotor and stator, and thus significant torque. Once the motor is at speed, the start capacitor is switched out of the circuit, and the run capacitor is left in the circuit to provide a smaller phase shift.