Impedance matching is used when one wants to pass the maximum amount of power (not the maximum voltage or the maximum current) from one stage to the next. You don't need to match impedances from source to load when you are trying to transfer a signal that is represented only by a voltage. In this case, it is best to have a low impedance source and a high impedance load so that you suffer the minimum amount of voltage drop in the source. This applies to the majority of experimenter circuits, generally speaking, including most logic circuits.
There are many times, however, when you do want to pass the maximum amount of power. For example, consider a stereo amplifier in your audio system. The transducer that creates the sound, that is the speaker, needs a lot of power to operate, and there really isn't any alternative to a speaker if you want to create a loud sound. Since it costs money to generate a lot of power, it pays to be efficient about it, hence our interest in getting as much power to the load as possible for a given amplifier circuit.
It is easiest to understand maximum power transfer by looking at the alternatives first. Let's say you made your load a very high resistance, while your source had a low output resistance. You would get a very good transfer of voltage, but not much current flow. Conversely, if you lowered your load resistance to a very low value, but left your source resistance higher, you would increase the current flow but your voltage delivered to the load would drop very low. Don't forget that power equals voltage times current so you want to find a way to have both of these be as high as possible. It just so happens that you get the maximum power when you deliver half of the available voltage and half the available current, which happens when your load resistance equals your source resistance. This is the point of maximum power transfer for resistive (non-reactive) sources and loads.
Note that I'm not using the word impedance here. This is important because it gets a bit more complicated when we say impedance. Impedance means the resistance plus reactance, a complex number, and reactance has a sign to it (that is, it can be negative or positive). I won't go through the math, but it is a fact that maximum power transfer occurs between a complex (meaning with resistance and reactance) source to a complex load when the resistance values are equal and when the reactance values are equal in magnitude but opposite in sign.
RF engineers get intimately familiar with this concept because at very high frequencies it is more convenient to measure power transfer than it is to try and measure instantaneous voltage or current at a particular point in the circuit. They learn quickly that impedance matching involves adding some capacitors and inductors between two stages to alter the impedance of a load so that it looks like the CONJUGATE of the source impedance. Conjugate is our big word meaning the same resistance and equal magnitude but opposite sign of reactance.