The extension says it needs:
1) Your data on dropbox.com and www.dropbox.com
2) All data on your computer and the websites you visit
#1 is obvious. I think it needs #2 so it can save a copy of the older version to disk and then launch the diff tool. If you want to check out the extension before installing it, see http://superuser.com/questions/337172/how-do-you-do-examine-...