Yeah just because it's free doesn't mean you don't have to DL and install it like any other game. So if you want to play it you're going to have to clear an adequate amount of space on your drive.
This should have been evident from the confirmation you got after adding it to your account, as it clearly says INSTALL YOUR NEW CONTENT, in all caps no less.
That said, it only requires a bit over 9GB space, which is tiny compared to most of today's games. Don't forget to allow for 15% unused space on your drive, and that's actual space, not the advertised capacity. This is advised by MS, as if you exceed 85% capacity, the OS and drive may not work properly.