CodeKit's Browser Refreshing feature wasn't working for me the other day, and the solution to my problem was not amongst the various Stack Overflow questions on the internet. So I'm posting it here in case anyone else has the same issue.
In the screenshot above, hitting the "Preview" button (top left), takes me to:
But that gives me a "Problem loading page" error. I realized that CodeKit was using my computer as-it-is-named within my local wireless network and that it might be conflicting with an old internet "Sharing" preference pane that I had installed years ago (which also uses my computer name on the local network).
As it turns out, in the "Sharing" preference pane:
Computers on your local network can access your computer at:
And the solution is to combine the two URLs. In other words, the URL where CodeKit's browser refreshing works for me is: