Do these things "work?"

Why do you think you need one?

That is, doesn't the nitrate have to turn into
something?!

Yes N2 and H2O via denitrifying bacteria present in your live rock.

See:
http://www.aquariumfish.com/aquarium...d=3793&search=

What is the waste product of the de nitrifying bacteria?

N2 and H2O.

Wouldn't the TDS levels still rise? If so, doesn't this mean water
changes are still vital?

Contrary to popular marine aquarium lore, you DO NOT NEED to do partial
water changes. My nitrites and nitrates are UNDETECTABLE using store
bought LFS test kits. I have not done a water change in over a year (very
close to two).

If you are the type of person that only changes water when the things you
test for change (rise in nitrates,
nitrites blah blah blah) then I can see this just disguising the need
to do water changes. Am I making any sense?

"...disguising the need to do water changes." ...hmm If something
causes your nitrites and nitrates to be very low than why would you NEED to
do a water change? Just for fun?

Of course I would never DREAM of slacking on my water changes (before
MG says something)

Stop following the herd!