This type of impedance matching maximizes voltage gain when input sources, voltage amplifiers and output loads are connected.
See attached PDF for diagram and formula.
You will see that the source impedance (\( R_s\) or \( R_{out}\) ) of a prior stage and the input impedance of the following stage (\( R_{in} \) or \( R_L\) ) form a voltage divider network that reduces the effective gain of the amplifiers (\( A_1, A_2 ... A_N\) ).
Note that different rules apply when you want to maximize power gain (transfer) and not voltage gain, as discussed here.