The back-slash (\) character is an escape character--meaning anything after it is interpreted as a symbol and not an operator or input separator or whatnot. So if you wanted to name a file ***IMPORTANT*** you could do a $>mv file \*\*\*IMPORTANT\*\*\* and that'd work. Without the \ in Linux\ Files, commands are looking for BOTH the directory /mnt/windows/files/Linux and the directory Files in whatever directory you currently are in. Another way you could accomplish what you're trying is to type $>cd /mnt/windows/files/"Linux Files" -- everything in the quotes is considered one "word" or file or directory. This gets nice as in when you're manually renaming long file names:
mv Artist\ With\ Spaces\ -\ CD\ Title\ -\ Track\ -\ Wrong\ Song\ Title.mp3 Artist\ With\ Spaces\ -\ CD\ Title\ -\ Track\ -\ Correct\ Song\ Title.mp3
mv "Artist With Spaces - CD Title - Track - Wrong Song Title.mp3" "Artist With Spaces - CD Title - Track - Correct Song Title.mp3"
It's just easier to read.