On a transmission line, why does max power transfer occur at impedance matching?
First start by reading here and and in particular the section "Single-source transmission line driving a load"

Why is max power is transferred when the characteristic impedance of a transmission line is equivalent to the impedance of a load.

Well that is not exactly true. You should say "equivalent to the Real part of the impedance of the load."

You should know that there are 3 passive elements: Resistors, Capacitors and Inductors.

Of those only the Resistor can dissipate power because it has a Real value impedance.

$$ Z_R = R $$

Capacitors and Inductors are reactive components and cannot dissipate power (we're talking about ideal components here). Their impedance only has an imaginary part

$$ Z_C = 1/jwC $$
$$ Z_L = jwL $$

That j  makes these imaginary.

These reactive componets can only influence the amplitude and phase relation of a signal. Since they cannot dissipate power no power is lost in these components.

An (ideal) transmission line can be seen as a distributed network of Capacitors and Inductors, so no resistors ! The characteristic impedance of a transmission line tells us something about the relations between amplitude, phase, currents and voltages of the waves traveling through it.

In the middle of a transmission line the wave traveling through it "sees" the same characteristic impedance in front and behind. It cannot dissipate into these impedances as they are reactive, they cannot dissipate power.

However at the end of the transmission line at the load, the characteristic impedance ends and turns into a real impedance. The amplitude and phase relations are not changed when the load impedance has the same value as the characteristic impedance of the transmission line. So the wave travels into the load as if nothing has changed. If there was a difference, then part of the wave would reflect.

In the load the wave cannot travel further but since the impedance is real it is dissipated and turned into heat.