I'm using Foundation 2010 and when I checkout various documents they won't save on my computer. I've tried to play around with the Office offline editing options and the Sharepoint Draft file location but it still doesn't work. This is on a Windows 8 Computer using Office 2013 and IE10.
It sounds like maybe you're misunderstanding the "Check Out" feature. If you have versioning options turned on for the document library, checking out a document using the web interface doesn't download it, it just marks it locked for editing by the curent user. You'll still need to hit the "Edit" link to download and edit the file after you check it out.
The idea is that you're putting a reservation on it and might save it several times as you make changes, then check it in, creating a new version.