7.1 Won't Install

This same problem happened to me, and what solved it for me was to forget trying to install the update through the Mac App Store. Just go to developer.apple.com/downloads and download Xcode 7.1 directly, then install it as if you were installing a beta version. Even though you haven't downloaded it from the Mac App Store, the "update available" notification should still go away if you check the Updates page.

Forgot to post this earlier: https://forums.developer.apple.com/thread/24089

Make sure you have enough disk space. If you install it through the app store after searching for "Xcode" rather than through the Updates interface it will tell you you are out of disk space. In the Updates tab it won't.