billcalley wrote:

We are all told that VSWR doesn't matter when using low loss
transmission lines, since the RF energy will travel from the
transmitter up to the mismatched antenna, where a certain amount of
this RF energy will reflect back towards the transmitter; after which
the RF will then reflect back up to the antenna -- where the energy is
eventually radiated after bouncing back and forth between the
transmitter and antenna.

As pointed out, VSWR does matter. A lot of bouncing means you heat the
transmission line with the power instead of radiating the power.
'Doesn't matter', really means it can be tolerated if need be.

I understand the concept, but what I don't
quite understand is why the reflected RF energy isn't simply absorbed
by the 50 ohm output of the transmitter after the first reflection?
For the RF to bounce back and forth, wouldn't the transmitter's
impedance have to be very, very high (or low) when the reflected RF
energy hit its output stages? I know I'm missing something vital
here...

Here is what you are missing. In the case of the output, (real/resistive
component of the transmitter), seeing the reflected wave, it is _not_
reflecting that power back up the transmission line as you think it is.
It would go back to that real impedance and heat the transmitter. Here
is what is done with a miss match in the real world.

trans-output - match - line - antenna

The 'match' is where the magic happens. All the energy coming down the
line that got reflected from the antenna 'sees' the 'trans-output -
match' as a perfect reflector and gets bounced back[*]. On the other
side of the match is the trans-output. There the trans-output sees a
perfect impedance, (technically, the conjugate of the trans-output), so
that all the power travels through the match toward the antenna.

The magic is that when the match is tuned, both of the above conditions
are satisfied.

*The reflected wave sees a purely reactive reflector not just because of
the network but also because of the output power of the transmitter.
Without transmitter power the impedance as seen from the load will
dramatically change.

Best, Dan.