There are lots of ways in which it could work:
Essentially, a media file would be encrypted, and you would be issued with a license file to go with it (no reason why the 2 can't be in the same file). The license file would include details on what operations are allowed, and a list of authorised equipment identified by serial number. This license would be digitally signed by the publisher using a signing key leased to them by the DRM chip manufacturer. Included in the license is the decryption key for the media. This decryption key would then be encrypted using the unique encryption key allocated to your DRM chip.
When you want to play a files, Software would send the file and the licence to the chip. The chip would verify the license is not a forgery, and would then use it's unique secret decryption key to obtain the real decryption key from the licence. As each chip has a different key, only the chip to which the license is locked can retrieve the media key from the licence.
The problem with this is that once the chip has done its work, software is free to access the decrypted data. If it's an encrypted audio file, then the software could then save the unencrypted data to disk.
There are a number of ways of getting around this - some manufacturers want to include a DRM chip in every output device (e.g. sound card, graphics card). E.g. you software sends the encrypted data direct to the sound card, and the sound comes out of the analogue outputs. The DRM chip could ensure that where licence conditions dictate the recording functions of the soundcard, and the digital out are disabled while protected media is playing. In this way there is no way for the software to access the unencrypted data. A similar technique could be applied to graphics cards for playing movies.
The overall goal with hardware DRM is to ensure that there are as few ways round the protection as possible. So that your PC motherboard would have a DRM chip - On boot this chip would verify the BIOS is legitiamte and that no unauthorised patches have been installed. The BIOS would then use the chip to verify that the OS is on an 'approved' list of DRM compatible OSs. If your OS is not DRM compatible then the BIOS will not be allowed to load it. Once the DRM secure OS is loaded, it will scan any program that is run to see if it is authorised to access DRM media. If the program is unauthorised then it not even be allowed to access the protected file. Indeed, the OS will have to save all DRM protected files onto a special partition on DRM enabled hard drives. If you took the HD out and attached it to a 'legacy' system, the drive would block access to the DRM partition.
In the early days, before all the safeguards are in place, there will be weaknesses - but gradually we will start seeing the barriers come up. We're seeing some now - some flash memory is now DRM enabled (Magicgate memory sticks, and SD cards) - authorised equipment/firmware has the option to accesss a secure partition on these cards, which is not available to normal readers.