Menagerie includes Exile mats, one per player. They're a place to put cards where they're still yours but not in your deck. The Exile mat has a built-in rule to let you get the cards off of it; check it out:
What that all means will vary a lot, depending on the card doing the exiling. Eight kingdom cards use Exiling, and six landscapes; here are some of those kingdom cards.
Bounty Hunter mainly stores cards that you probably don't want to draw anymore. If you didn't have one of whatever it is in Exile already, you get +$3, so early on it helps you spike $ amounts, and then it's either a way to at least get rid of more cards, or you can consider collecting a bounty on something you weren't exactly planning on parting with. And after that it can keep Provinces out of your deck, so I mean, what's not to like.
Stockpile has a different trick: it Exiles itself. You can just buy another one to get back the first one, and Exile them again, and buy another to get back those two, and so on. Until the pile runs out anyway.
Coven has yet another trick. It hands out Curses, but they don't just poison your deck immediately; they accumulate in Exile and then one day you dump them all into your deck at once. I will just tell you now, you better not be relying on Bounty Hunter to get rid of the Curses; you can keep collecting the bounty, but Coven will keep dumping them back into your deck.