Downloadable content is licensed to both your gamertag profile and the Xbox 360 console it was originally purchased on. In order to play DLC offline or on other profiles on the same system, you will need to use the same console that the DLC was originally purchased with. DLC content may appear as corrupted when attempting to access it on a different system if you are not logged in to Xbox LIVE with the gamertag that purchased it.
If you have changed systems or had your Xbox 360 replaced, you can perform a license transfer to move your licenses to the new system. Without attaching your licenses to your current console, you will only be able to access the downloadable content when logged in to the profile it was purchased with and when connected to Xbox LIVE. For more information on this process and DLC licenses, please visit this Xbox support article: http://support.xbox.com/en-US/xbox-live/marketplace-and-purchasing/download-content