What that means is, you work with the files as if they were completely local to your computer, since they are. Dropbox monitors for changes, and makes sure to update all the other copies of these files. But you always get the speed (and convenience) of working with local files.
That also means that what the parent says is right - you have extra copies of your files lying around on any computer that is connected to Dropbox. These are full physical copies of the latest version of each file.
Of course, if there is some bug that makes Dropbox think all your files were deleted or something, that change could propagate to all your other computers at once and delete all your backups. I agree that it's not very likely.
Not very likely, and if you've something like OSX's Time Machine it'd be trivial to go back.