Please provide an intuitive explanation of maximum power transfer.

How can same source impedance and load impedance provide maximum power transfer?

Please provide an intuitive explanation of maximum power transfer.

This is a conceptual question therefore I want to ask with an example. I'm wondering in which cases impedance matching is used.

For example, if one has a sensor which can be modeled as voltage source call it Vs. Imagine it is outputting mV level voltage signals and it has 1k output resistance. And if we want to amplify it we would use an amplifier with a very big input impedance like 10Meg so that most of the Vs reach the amplifier input.

But for maximum power transfer the input impedance of the amplifier must be equal to 1k which would halve the Vs reaching to the amplifier input.

If we have a signal to be amplified why would we match the source output impedance to receiver's input impedance? In my example it was not a good idea. And I know that in radio transmission impedance matching is used. But again I'm stuck at the same point. There must be a difference in two cases where we need maximum voltage at the receiver or we need maximum power at the receiver.

Can this be explained by two examples for each case?