Hiya, apple0134!
The folder in question would've been created after the repair of a particular issue with the structure of your hard drive. I would very strongly recommend using Disk Utility to verify the drive and repair it as necessary.
Disk Utility 12.x: Repair a non-startup disk
http://support.apple.com/kb/PH5863
- Close files and quit all applications on the disk you want to repair.
- Open Disk Utility, in the Utilities folder in Launchpad.
- In the list at the left, select the item you want to repair. (Be sure to select an item that’s indented to the right in the list, not an item at the far left.)
- Click First Aid.
- If Disk Utility tells you the disk is about to fail, back it up and replace it. You can’t repair it.
- Click Repair Disk.
Regarding the 'DamagedFiles' folder, this is typically created when an 'overlapped extent allocation' error occurs on the drive. The folder itself doesn't contain any actual files. What you see in the folder is only a pointer to an actual existing file. I would make note of where those files exist and verify that the actual original files are functional. To easily locate the original item that an alias points at:
OS X Mountain Lion: Create and use aliases
http://support.apple.com/kb/PH10609
You can find the original for an alias you added to a Finder window or the desktop.
Select the alias, and choose File > Show Original or press Command (⌘)–R.
Also, more information regarding the 'overlapped extent allocation' error:
Handling "overlapped extent allocation" errors reported by Disk Utility or fsck
http://support.apple.com/kb/TA22517
Cheers,
Allen