Why does this common-source MOSFET amplifier work?
The anode-cathode current for the GU-81M is modulated by the FET. All the current from the valve flows through the FET and it's DC configuration (using the potentiometer) will set the operating point cathode voltage.   The center tapped heater has no influence and the heater/cathode is not at zero volts. 

Yes, this would be considered cascode operation as in this audio amplifier with two triodes: 

 

In your case since the FET modulates the operating current (and therefore the C-G) of the pentode it changes the amplitude of the RF output.

If you look at the datasheet for your FET:

 

Excuse my mouse scribbling on top of the graph, but here you can see as the Vgs varies, the current through the valve will vary (and the cathode voltage).

I've no idea where you operating point for the GU-81m is so cannot work out the valve current, but you should be able to. 

Note that your grids are at very low voltages, so as the cathode rises they are effectively negative biased. It's doubtful you'd see more than 120-150 V on the cathode so well with the ratings of the FET.