Why can't I install this downloaded APK? I downloaded an APK and when I try to install it, the "install" button is completely unresponsive.

I have gone into the settings and enabled "Unknown sources" in the security section.

Is there something else I need to do?

I'm using a Galaxy S5 with Kitkat 4.4.2, unrooted.
Disable your screen dimming app. :)

Some screen dimming apps work by overlaying the whole screen with a translucent window, and letting touch events pass through to the activity below, but for security reasons, you can't interact with system dialogs (such as the package installer) through another activity.

It sounds counter-intuitive, but it could be the cause. It's more likely than a corrupt APK file or a version incompatibility, either of which would cause an error message.