hi,
I think where people are getting hung upon is the phrase maximum power transfer.
Maximum power transfer means transferring the maximum available power from the source to the load.
eg:
If a Voltage source of say 10V with an internal resistance of say 10R, this means the maximum available power is 10Watts.
So to transfer the maximum amount of power to the load, the load resistance has to be 10R.
So when the load resistance is equal to the source resistance, in this case, the total loop resistance is 20R.
This means that 0.5A flows [exists] in the loop, so the power dissipated in the source and load resistance equals 2.5Watts each.
Thats a total of 5Watts from a possible 10W, so the transfer is only 50% efficient.
If the Load resistance is lower than the source resistance more power power will be dissipated by the source resistance
If the Load resistance is higher than the source resistance less current will flow in the loop so the power dissipated by the the load will be lower
The above DC, resistive, explanation should answer the OP question
To maximise the power dissipation in a load, the source resistance should be lower than the load resistance.
If you dont agree, consider the low output impedance of an audio amp, say 0.02R driving a 4R speaker.