Happy gaming, and enjoy your new DLC content!
Behind the scenes, Sony hosts these .pkg files on their servers, making them available for download. When a user purchases DLC, the PS3 sends a request to Sony's servers, which then deliver the .pkg file to the user's console.
The installation process of DLC involves the PS3 reading the .pkg file and performing a series of checks to ensure it's valid and meant for the user's console. The file is then decrypted and installed into the appropriate directory on the PS3's hard drive.
Launch your game. You should see a notification like "Downloadable content detected" or new items in your inventory.
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