You're absolutely right that a rotor inside a magnet on a single-phase circuit won't start moving on its own. The key to getting it started is having a rotating magnetic field. The field of a magnet in a single-phase circuit doesn't rotate; it just switches polarity. Once the rotor is turning that's all it needs, but when it's stationary it needs a push to get it moving.
There are several ways to create that push. Many motors have an additional winding, called a starter winding, that's offset from the main winding. The power to that winding goes through a capacitor (that's the thing inside that cylinder on the top of the motor), which shifts the phase of the power coming into the starter winding. So as the incoming voltage rises to its peak, the starter winding produces its full magnetic field, and then, a bit later, the main winding hits its peak. The net magnetic field changes its direction as the main winding kicks in. That pulls the rotor a bit. Then the voltage switches direction, the momentum of the rotor keeps it moving, and the pull in the opposite direction pulls it around. Again, the starter winding gives the magnetic field a bit of rotation, and that, again, pulls the rotor more.
Once the rotor is up to speed, a centrifugal switch shuts off the starter winding, and the main winding is left on its own to provide all the power.
As you put more load on the motor, the rotor slows down, and lags further behind the magnetic field. That produces a stronger pull, which produces more torque, which adjusts for the increased load on the motor.