Basically, let's say you beat the final boss on Level 34 PoE, he drops a Treasure Key and after, you go into a room with a big chest. You open the chest using the key that just dropped for you and you get you're rewards (Legendaries, Exotics, Etheric Light etc). Now let's say you run Level 34 PoE again, when you beat the boss a 2nd time he won't drop a Treasure key this time, however if you have one in your inventory, you can use that key to open the treasure chest at the end when you get to it.
In case you're wondering, you can only use one key per run. You can't use all 5 keys at the same chest
So yeah, farm the hell out of those keys. You'll need them.