I’ll include below, verbatim, the answers that Eric Lawrence (creator of Fiddler) kindly provided on the Fiddler forum:
One possibility is that your computer is configured with an Intranet zone and that Intranet zone is dependent on a proxy configuration script: http://blogs.msdn.com/b/ieinternals/archive/2012/06/05/the-local-intranet-security-zone.aspx. When Fiddler is running, the proxy settings are pointed at Fiddler itself.
… there’s another factor at work here if you’re using an Intranet site as the target of an XHR from a site in the Internet zone.
Edge runs in Enhanced Protected Mode (AppContainer). That has a feature which blocks access to Private Network Resources from Internet-Zone processes. See the “Private Network resources” section of http://blogs.msdn.com/b/ieinternals/archive/2012/03/23/understanding-ie10-enhanced-protected-mode-network-security-addons-cookies-metro-desktop.aspx for more details.
I added local.myapp.test (the URL I’m running my SPA from) to the Local Intranet zone in Internet Options and now Edge is happy without the need for Fiddler.