I rented the original game. I'm guessing I need the disc to play this?
Sadly, yeah, you will need the original disc. In order to play Burial at Sea episode 1, you have to put the actual Bioshock Infinite disc in and buy Burial at Sea from the Bioshock: infinite main menu (make sure your PS3 is connected to the internet before doing so). And afterwards, you still have to put the disc in the PS3 in order to play Burial at Sea (which you access from Bioshock Infinite's main menu).