How can I recover the files in LOST.DIR on my SD card, and what are they?
According to this Super User answer, LOST.DIR is the folder  mac disk repair looks for as part of it's repair process.

LOST.DIR is something mac disk repair looks for and will ask about as part of it's repair process. If it's related at all to the standard lost+found, then it's a special file that holds all the corrupted or orphaned files found during a file system check.
Something is checking your drive and finding a problem it wants to make note of in LOST.DIR. I think this sort of thing is more likely to happen with Macs on non-HFS drives.


Android also creates this directory as part of it's filesystem check and puts files there which Android believes are corrupt.