TL;DR: If a game comes with "DLC Keys", you must register those keys, before you will be given access to the DLC. Furthermore, if the key has been used, it will likely not work again. This should only be an issue with pre-owned games.
You report not having entered the keys that came with your copy of Destiny, yet.
When a retail version of a game comes with DLC content, it is most common for this DLC to be provided in the form of a DLC key. The idea is that you enter the code on the related store screen, which registers the DLC to your account.
As DLC can be run directly from the hard drive, and does not require a disc, this gives the developers the security of knowing you will not simply install the DLC, and exchange the disc for something else.
In other words, the game is telling you that you have to purchase the DLCs, because you simply have not registered them yet. As far as Sony are concerned, you have not purchased the rights to play them, yet. Once you have registered the keys with the PlayStation store, Destiny will acknowledge DLC access, and you should not have any more problems.