A single phase motor is nothing but an AC electro magnet. In order to spin the rotor we need a rotating electric field. By adding two more coils and a capacitor to cause phase change we get two magnets and a time delay between their cycles - like magic we have a rotating field.
Many parameters of inductance, resistance and capacitance need to be optimized for high efficiency. Once the motor is up close to synchronous speed some designs switch out the second set of coils with a centrifugal switch. Now the secret is the circulating current in the squirrel cage rotor which turns the rotor into a second magnet.
Normally a run capacitor is used in a Cap Start, Cap run type or cap start & run type single phase induction motors which are designed to share the load between a main winding and an auxiliary winding having series connected capacitor. These motors are not to be compared to a cap start, induction run or resistive start, inductive run type motors since they use auxiliary winding for start purpose only. The run capacitor in first types of motors contributes towards improving motor power factor and possible reduction of line current drawn and thereby the input power drawn as compared to the second type of motors which are basically a low power factor type motors consuming higher input current & thereby higher input power. More theoretical / mathematical proof can be found in any standard single phase induction motor design book if one is really interested to learn.
Also note the answer to second part of the question as to what happens if run capacitor is removed is that motor load carrying capacity will be halved almost and in some cases it can burn the motor windings if the load is not a fan type load but constant inertial type.