Why does a motor or compressor need a capacitor to run properly?

it is used to create magnetic field in order to attain rotating torque.

There are two common types of motor capacitors, run capacitors and start capacitors:

Run capacitor to energize the second-phase winding (auxiliary coil) to create a rotating magnetic field while the motor is running

Start capacitors briefly increase motor starting torque and allow a motor to be cycled on and off rapidly