The automatic HF ATU I use (yes, another one of those darn hams) has around twenty latching relays that switch in various configurations of coils and capacitors while a microprocessor tests the quality of the match with the antenna. Happens in about half a second, and sounds like a bunch of angry bees have temporarily woken up.
In theory, this can be a continual process that adjusts itself automatically without any messing around with buttons. However, the automatic matching circuit in most gear does need a steady burst of radio frequency energy to be transmitted so it can sniff what's happening and voice is just too bumpy on HF SSB for that (although smarter algorithms should cope). Hence the special button that squirts out nice solid carrier.
One analogy is that it's like a gearbox; you want to run a car engine within its most efficient range of revs no matter what the speed of the car is, so you find the right ratio to couple the engine to the wheels for the particular circumstances. That way, you get the most power out of the thing no matter what the speed. Alternatively, try going at 80mph in first gear and something will break! In the old days, Sparks had to do all the matching manually, which can be quite a complex business on valve gear where you have to watch quite a few parameters (including some that can generate plenty of real sparks), but now we've gone through the synchromesh stage to the fully automatic.
A mismatched antenna will still work, just less efficiently and - as has been said - a disasterously mismatched antenna will reflect so much power back at the amplifying transistors in the radio that they'll fry (although most modern radios should detect this condition and just grumpily shut down or reduce power while flagging their unhappiness).
It is theoretically possible to check the quality of the antenna match without transmitting (which isn't the most social thing to do) by making it one leg in an impedance bridge and tweaking for peak noise in the receiver, but that seems to have dropped out of fashion.
R