There are two ways:
1. software hack with USB patchstick to add SSH, which will allow you to login into the ATV with Cyberduck app or similar with password "frontrow."
2. Open the puppy up, pull the hard drive, attach it to a "universal drive adaptor" that connects to the hard drive on one end, and USB into your comp on the other.
As an example if you chose (2) when you plug in the ATV drive into your comp you will see OSboot and Media partitions mount. Go into Media and your files will live in /mnt/Media.
Unfortunately if your video files are not tagged with metadata it will be a real pain because the files on the ATV have random looking names and will be sorted into dozens of nondescript folders.
Neither option is really that hard, but you would need to do some googling to get all the steps software wise or be comfortable opening the ATV.