A stationary single phase motor is like a car trying to take off in 4th gear from a stop light. AC voltage has in-phase current and voltage cycles at 60 per second. With a capacitor added to the circuit, the current is phase shifted 90 degrees ahead of the voltage (it leads the voltage by 90 degrees). A motor is an inductor so a quick current surge creates torque (think automobile starter motors, they are especially wound for current to develop torque). So the compressor motor gets a very quick torque "kick-in-the-rump" to get it spinning. The voltage follows 90 degrees later.
If the start capacitor is too small (not enough capacitance) or defective, or if the load is too great, such as a bad bearing, compressor stuck, starting under pressure (unloader not working), the motor can't spin. But for now let's say all that is good.
If you turn the breaker on and the compressor main switch (on/off switch) is on and the tank has no pressure and the compressor doesn't run, it may not have voltage applied to the motor. Bad pressure switch, bad start relay, bad on/off switch or loose wiring in the power plug or power connection.
