There are a few parts to this, it's good to keep them separate.
1. Antenna impedance
Antennas are complicated beasts, they connect the circuit world of voltage, current and power, into the electromagnetic world of Electric and Magnetic fields, and EM waves.
The antenna impedance is made up of the radiation resistance, loss resistance, standing and travelling waves on the antenna, reflections from nearby objects, the matching circuit, interaction with the ham holding the mast up, etc. All of these things change quickly with frequency.
But for this discussion, all you need is that at some frequency, the antenna has a connector or terminals, and looks like an RL or RC circuit. Its impedance is fixed and is expressed as R+jX. (Real and Imaginary parts).
2. Standing waves
Are an effect on transmission lines. When the termination impedance is not the same as the characteristic impedance, the travelling waves are reflected by the termination, This causes standing waves, places where the magnitude of the RF voltage is larger and smaller. This effect is stationary on the line (because the phase relationship between the forward and reverse waves is fixed).
The standing waves themselves have only one interesting effect: they lead to localised heating in regions where the current is larger (and sparking will happen first at the voltage maxima).
The presence of the reflected wave also means that the impedance looking into the start of the line is not Z0 but something else, and there are formulas for that.
We often describe the degree of mismatch by the Voltage Standing Wave Ratio. For example:
- on a 50 ohm line
- with a terminating impedance of 25 ohms
- VSWR will be 2:1
- after some length of transmission line, impedance will be 25 ohms, 100 ohms, or any complex impedance between these.
A transmitter is (for now) a circuit of lumped components (transistors, resistors, capacitors etc, but all obeying normal circuit rules), designed to with a 50 ohm load at the antenna connector. It will also work with 40 ohms or 60 ohms, but may not work with 1 ohm or 10k ohms.
The question is about what impedance variation is OK, and how the amplifier might be damaged outside of that range.
One can certainly buy amplifiers that are safe and work without "cutting back" or self-protection, into any impedance. But these are not optimal for most users. This one is 10 watts out, consumes 400 watts, weighs 28 kg, and probably costs $50k.
So the designer optimises the amplifier for power and efficiency, in limited frequency bands, and one of the results is that the amplifier can't be operated at full power, with any impedance load. A short or open circuit will damage it.
This happens because the final transistors in the amplifier have a maximum voltage, current, and power dissipation, and operating them out of this envelope will quickly damage them. Of course you could get bigger transistors, but they're more expensive, larger, have more capacitance, so the designer makes a reasonable choice based partly on the range of load impedance that is expected. So an amplifier that delivers full power at better than 2:1, will be cheaper and smaller than one that tolerates 4:1.
Because transistors will fail almost instantly, there are usually protection circuits to detect these conditions and quickly reduce or trip the power, so it's safe but not really transmitting any more.
An aside - there is no 50 ohm series resistor in the amplifier, in fact its output impedance is probably much less than 50 ohms. This means that it can probably tolerate high impedances better than low impedances... VSWR is a very blunt measure of impedance, but it's easy to measure. Some discussion in this answer.
How these all come together
In practice, the user starts with (1) a somewhat mismatched antenna. Connects a long transmission line to it, which changes the impedance (2). Connects this to the amplifier (3), which sees the transformed impedance at its terminals.
The question is whether it's OK with this impedance, or it's too far from the design value and it is damaged (or cuts back).
I think it's important to frame it like this, not as "reflected power coming back and damaging the amplifier". The amplifier is self-destructing when connected to the wrong load impedance. The load happens to be an interesting electrically-large circuit.