Automatic voltage stabilizer with isolation transformer
Unfortunately, if you don't have the control circuit, you don't have a voltage stabilizer.  This is not anything like a constant-voltage (or more properly, a ferroresonant) transformer where voltage stabilization is achieved through purely passive means inherent to the construction (and LC tank) of the transformer itself.
All you have is a motor-controlled variac.  It stopped being a voltage stabilizer the moment the control board was disconnected.  It would be like saying you have a car because you have 4 wheels in your garage.  Yes, you have the part that actually touches the road, does the rolling, but no one would call 4 free-standing wheels a car.
When you say you powered it up and it worked, well, no it didn't.  You can roll a wheel down hill, that doesn't make it a working car.  This will of course output a voltage if you power it up, but it isn't going to stabilize anything.  The voltage you get out will vary with the input voltage and with the load just like any other transformer.  So if by 'works' you mean 'is no different than a plain old transformer and completely unable to perform the very function it is named for', then yes, it works.  And if you just want an isolated variac that has a power feed (as in, motorized actuation, not power like electrical power), then great, that's exactly what you have.  That is still something pretty handy to have! But if you wanted a voltage stabilizer, I'm afraid that was the control board, not the thing it controlled.
As for how what you have works, it is just a transformer constructed similar to a variac.
A variac is an autotransformer (which has only one winding that serves as both primary and secondary) combined with a mechanical and a wiper, much like inside a potentiometer, that moves across a flat surface of exposed/uninsulated sections of the transformer winding, effectively allowing you to switch between many many taps on that single winding by connecting to the winding through the wiper (which typically has a conductive graphite brush on the tip).
Like this:  
Imagine a normal transformer with an open secondary and energized primary.  The primary has a center tap, but you are powering it across the entire primary, the tap is not connected.  If you were to measure the voltage at center tap on the energized primary, you'd find half the input voltage that is across the entire primary winding.  This voltage varies linearly with with the tap position along the coil.
Well, you can ditch the secondary entirely, and adjust your output voltage by just tapping off a single winding.  That is an autotransformer.
Combine this with the many-tap ability of the wiper and graphite bushing and you can vary or switch between sometimes hundreds of 'taps' (just a point along the winding) mechanically.  Usually with a knob, but it can be motorized as well.
What you have is a just this.  A variac/autotransformer with the motor that can move the graphite brush.
What I previously described has the downside of not being isolated however, but adding isolation is simply done:  have a primary, then do the same thing we were doing with the variac's single winding, but with the secondary winding.  Its just a secondary with many many taps that can be mechanically switched between, in this case by a motor.
However, you may want to double check that, as the circuit you've provided will not have any isolation.  Either that diagram is incorrect, or the isolation occurs somewhere not shown in the diagram, or what you have isn't actually isolated, so you might want to double check.