Ammonium thioglycolate is the ammonium salt of thioglycolic acid. Having a thiol (R-SH), thioglycolic acid is a decent reducing agent, particularly of disulfide bonds (R-S-S-R’).
Your hair is cross-linked by legion disulfide bonds; determining whether it’s curly or straight. Thanks to some chemistry, you can rearrange that frizzy hair. You can reduce it with some thiol (in this case the less-stinky thioglycolate; complain all you want about the stink, but be glad nobody’s using beta-mercaptoethanol, which would work just as well, at the cost of all your friends). Style it and oxidize it back with some hydrogen peroxide and you’ve got your disulfides right where you want ’em, from foofy ’80’s hair, to stock-straight locks.
It always bothered me that Reese Witherspoon mispronounced the chemical so badly in whichever Legally Blonde movie it was. She went to Stanford! She’s descended from John Witherspoon! Both her parents work in medicine!