The basic idea is that you can counter your opponent's spells and abilities with it. I think the idea you're missing is that any ability on a card acts at instant speed (just like an instant spell) so you can use it anytime you have priority (unless the card specifically states otherwise). So the idea is that you have to use the ability in response to them using a powerful spell to guarantee you get some value.
Learning what priority is and how to use it is a little complicated but the basic idea is whenever a player does something their opponent gets to react. Though how you react is restricted, usually only instant spells or unrestricted abilities actually work as ways to surprise or counter your opponent.
So, here's an example:
- I cast Hero's Downfall targeting your Sphinx
- In response, you activate the second ability, giving it hexproof.
- Hero's Downfall tries to resolve, but it now has an illegal target (due to it being hexproof) and is countered.
Congratulations, you've discarded your worst card in your hand to counter a very effective removal spell. Not a bad trade. The real power is you can keep doing this as long as you have a card to discard. So if I have 4 removal spells but you have 4 lands in your hand then I cannot kill your sphinx.
The only real downside here is if I'm an aggro deck I can use those spells to force you to tap him so you can't block. That strategy only buys me 1 turn per spell though, so it's not very good for me.