The relatively simple business of how valves/tubes work isn’t that well treated on the net, so I’ll have a go. I’m not so hot on physics so I will be very happy to be corrected on any of this.
The basic principle is that electricity can flow in a vacuum or a near-vacuum. That’s what’s happening inside a flourescent light tube: the electricity – a stream of electrons – flows from one end to another and lights up some gas as it goes. Two poles, one at each end, are connected to the mains, and the current flows through the tube from one end to another.
Basically there are three of these wireable-uppable poles inside a valve of the type used for amplification. One (the anode or plate) is connected to a source of electricity, and a second one is earthed (the cathode). The electricity wants to flow (a bit like lightning) from the plate to earth* and it would do so if it wasn’t for the third item in there, which sits between the plate and the cathode and is used to limit the current flow. This is the grid.
The grid needs to have some negative voltage on it to limit the current flowing through the valve. That’s why valves are called valves – it’s like a water pipe with a tap on it – the grid is the tap and it limits the flow going from one end of the pipe to the other.
The grid can only limit the current flow if there is negative voltage on it. The more negative volts, the more the current is limited. More negative volts on the grid is like turning the tap down.
This negative voltage is the ‘bias’ voltage. It sets a standard current flow. Without this bias voltage the current would flow too much, overheating the plates (they then glow red hot) and destroying the valve.
OK now here is how amplification happens. We also put the electrical waves that comprise the sound ‘signal’ on the grid. So the negative grid voltage is constantly varying with the sound of the guitar. That means that the current flow through the valve is also being varied exactly in tune with the variation on the grid. This means that the high positive voltage on the plate is varying too, many many times a second. The volts build up on the plate when the grid voltage limits the current, and lessen when the grid allows more current to flow.
This is a speed-of-electricity process, easily fast enough to follow sound frequencies, and even radio frequencies in fact. The voltage on the plate is varying fast enough to follow exactly the variation fed into the grid, echoing the sound waves precisely – current flow is lightning-fast and the effect of the grid variation is equally fast.
So there is a small voltage wave on the grid – and a much bigger voltage wave on the plate. That’s valve amplification.
*In fact electrons flow the other way, ‘upwards’ from earth. So does lightning by the way. But by convention the direction goes the other way, because all the circuit conventions were well established before the true direction of electron flow was discovered. So the cathode is heated to make the electrons flow from it, and the positively-charged (relative to the cathode that is) plate kind of sucks up those electrons. And the cathode isn’t always directly earthed, it just has to be negatively charged relatively to the plate… but you don’t need to know any of this to get the basic drift.