How does a diode valve work based on this sentence?
The statement is a bit misleading. In vacuum tube diode the cathode is heated but the anode is not. That means the cathode emits electrons but the anode does not.

So if you apply a positive voltage to the anode and a negative voltage to the cathode the electrons emitted by the cathode move towards the anode and a current flows.

However if you apply a negative voltage to the anode and a positive voltage to the cathode no current flows because the (unheated) anode doesn't emit any electrons.

The comparison with a transistor seems odd, because a transistor is not like a diode. It's more like a triode. The obvious comparison is with a semiconductor diode, though this works by a completely different mechanism.