When the heater is switched on the cathode heats up. Electrons in the metal gain enough energy to escape from the surface and are then attracted across to the anode. (The electrons are shown in this animation as small charged runners!) The hotter the heater the more electrons escape and the greater the current in the anode. If there was any gas in the diode the electrons would collide with the gas molecules and the current through the diode would be much less.
Reversing the polarity of the anode and cathode plates would reduce the current to zero. Increasing the anode voltage would increase the speed of the emitted electrons but not their number.