Several users have noticed a problem where their Mac's screen will shut off when they're either downloading or viewing video files. This will sometimes happen when the videos are viewed, but other times people have noticed it when just copying video files.
Specifically, this problem seems to happen with videos encoded in the FLV format, which is the container format for delivering videos through Adobe's Flash player. Though it is bizarre that copying the files leads to this issue, perhaps the loading of video previews during the copy process may lead to the problem.
If this problem happens, you can usually get your screen back using either a system feature or third-party application that resets the display (changing resolution, color depth, or other display properties). This can be a screensaver or Front Row, which can be set up to toggle with hot-corners or a key combination (Front Row is by default activated with Command-Escape). Additionally, you can force the system to reset the video output by pressing the Control-Shift-Eject key sequence or by unplugging an external display and plugging it back in again.
Since this behavior is specific to Flash content, there may be a problem or conflict with the computer's Flash installation, and resetting it should hopefully clear the issue. First, try simply reinstalling Flash (the latest version can be found here), but if that does not work, try removing the following Flash-related resources in your system and then reinstalling:
/username/Library/Preferences/Macromedia/Flash Player (folder)
/username/Library/Preferences/Flash Player (folder)
/username/Library/Caches/Adobe/Flash Player (folder)
/Macintosh HD/Library/Internet Plug-Ins/Flash Player.plugin
/Macintosh HD/Library/Internet Plug-Ins/flashplayer.xpt
In addition to removing and reinstalling Flash, you may try using another media player to handle FLV videos. A few options include MPlayer OS X Extended, VLC Media Player, and the Adobe Media Player. Once installed, set the system to use these players by getting information on an FLV video file, choosing the preferred application in the "Open With" section, and then clicking the "Change All..." button.