When trying to use the Edit Page feature of SharePoint, the page refreshes but does not change into edit mode.
This issue occurs when you are using Internet Explorer 11 on a server without the latest ASP.NET fixes applied.
There are two fixes:
Client side: Add the site to your compatibility view list by clicking the settings icon in the top right of your browser and selecting Compatibility View settings. Click Add to add the website to the list.
Server side: Update to the latest ASP.NET and SharePoint fixes.