I'll take a stab at it with the little I know. Once the switch is flipped there will be a rotating magnetic field in the stator (more just flip flops in a single phase motor, doesn't truly rotate unless it's a multiphase motor like a 3 phase), this magnetic field rotating in the stator (the stator is the windings on the outside i.e. that encircles the rotating bit in the middle) will rotate according to the frequency of the voltage, 60hz (or times per second) in this country. That comes out to 3600RPM for a 2 pole motor (1800RPM motors are 4 pole).
When you hit the switch there is instantly a magnetic field in the stator that is rotating at 3600RPM. There is nothing mechanical rotating in the motor at this point. Only a magnetic field that you can't see. And the stator NEVER rotates, it's fixed to the motor housing, all the stator ever does is creates this rotating magnetic field. Inside the stator and also inside the stator's rotating magnetic field is the rotor. The part of the motor that spins. It also has windings but these windings are not connected to input voltage like the stator windings are. What these windings are for is to accept energy from from the stators magnetic field, they do that through induction.
Induction- if you were to have a magnet in your hand and you were to move it over a wire it would INDUCE a voltage in that wire. You can't just hold the magnet near the wire and get current, you must move it. But if you do (you can also move the wire and leave the magnetic standing still, same effect) then that movement makes a voltage (or a current in a closed circuit). The faster you the relative movement between the magnet and wire and the greater the current.
Back to the motor. You have a magnetic field in the stator that is now rotating at 3600RPM (60 cycles per second times 60 seconds = 3600RPM) and the rotor, which is not moving, sitting in the middle of this field. The relative movement between this rotating magnetic field (always 3600 RPM) and the rotor (0 RPM) induces a large current, much larger than the current listed on the motor spec plate, in the windings of the rotor. With a large current now in the windings of the rotor the rotor will also have it's own magnetic field (current flowing in a wire makes a magnetic field). And the magnetic field of the rotor will be a strong one because there is a large current in it's windings. The magnetic field of the rotor rotates at the same speed as whatever the rotor is spinning at (which is still zero right now, right after the switch was flipped) This magnetic field in the rotor wants to synchronize with the stators magnetic field and that causes the rotor to begin turning, to mechanically accelerate. The rotor wants to be at 3600 RPM. The stator's magnetic field is always 3600 RPM even though the stator itself is stationary, the rotor's magnetic field is fixed to whatever the rotor itself is spinning so in order for the two to try to synchronize the rotor must accelerate.
When the rotor starts to accelerate the speed difference between the magnetic fields in the stator and rotor will get smaller. For example when the rotor has accelerated to 2400RPM there is now only a 1200RPM difference between the rotor and stator's rotating magnetic field. Because how much current in induced in the rotor is dependent on this speed difference (more speed equals more current) this smaller speed difference causes a smaller current to be induced in the rotor and thus a weaker magnetic field in the rotor. This weaker magnetic field means less actual work going into turning the rotor. As the rotor keeps accelerating eventually you reach a point of equilibrium where this magnetic field is so weak it's only creating enough power to turn the rotor against whatever little resistance exists in the motor itself (bearings).
This is why motors will say their no load RPM is something like 3500 instead of 3600 RPM. There needs to be a 100 RPM speed difference between the rotor speed and thus the rotor's magnetic field speed and the stators permanent 3600 RPM magnetic field speed in order to make enough power to just overcome the resistance in the bearings. As you load the motor the rotor speed (the output shaft is connected to the rotor) will decrease thus causing a greater speed differential, thus cause more current induced in the rotor, which cause a stronger magnetic field in the rotor, which causes the two magnetic fields to work harder again to synchronize (or speed back up).
So if the stator's magnetic field is rotating at 3600 RPM and the rotor (and thus it's magnetic field) is rotating at 3600 RPM there is NO induction of current in the rotor, NO magnetic field in the rotor, and NO power accelerating the rotor anymore. This is impossible because there will always be some power needed to turn the rotor, even against no load so the maximum rotor speed is a little less than this by some bit. On the other hand when the stator is receiving power from the wall and thus it's magnetic field is rotating at 3600 RPM but the rotor is a zero speed there is MAXIMUM current induced in the rotor, MAXIMUM strength of magnetic field, and the rotor goes zipping up to speed. If the rotor happens to be locked up on something and can not turn then when you flip the switch the motor is locked into this maximum current situation. Big currents are induced and things stay that way. The normal thing is for this MAX induction situation to only last maybe a second as the rotor is zipped up to speed quickly when you flip the switch. But if the motor is locked up that doesn't happen.
As far as I know that layman explanation is correct but some EE can refine it or debunk it.