Draw XY plane.
Apply a sinusoidal signal controlling the position of your pen in X direction. You have a pulsating force. It won't rotate. It can maintain crappy, torque-ripple-y rotation if this rotation is started somehow. It's like a single cylinder engine.
Apply another sinusoidal signal controlling the position of your pen in Y direction. Again you have a pulsating force but this time, perpendicular to the previous pulsating force.
Now put your pen on the paper and start drawing. You are drawing a circle! It's rotating!
This is called 2-phase electricity (and has nothing to do with the split-phase system of the U.S.A., sometimes incorrectly and confusingly called 2-phase), and it can make 2-phase motors rotate; these motors have two sets of winding, 90 electrical degrees apart. Like a 2-cylinder engine.
Pure 2-phase motors still exist and are driven by inverter/variable frequency circuits, examples include BLDC fan motors and steppers.
Your "1-phase motor" with a start and/or run cap is also basically a 2-phase motor. Kinda. More on that later.
2-phase electricity has never been in wide use for a long time. In residential power, we have only one phase. So we need to somehow generate the 2nd phase to run 2-phase motors.
A capacitor together with the inductance of motor windings introduces phase shift and builds this second phase. Because a capacitor stores energy, it charges from the mains, and supplies this power into the circuit later.
It's impossible to get 90 degrees apart, it will be less, so the motor can be mechanically designed to not expect 90 degree phase difference either; so it's a compromise, but it works. This secondary phase usually doesn't provide as much torque as the "main" non-delayed phase, but enough to make the motor start, and in the case of run cap, increase torque and reduce torque ripple.
Sometimes both start and run cap are used, to provide highest possible torque at start, but then reduce the effect of this secondary phase to prevent overheating it, without completely getting rid of it (which happens when only using start cap).