The inability to use Split View is typically because the user has upgraded OS X from a prior release of system software, and a particular setting has carried forward which is preventing Split View from working. But not to worry, this is a very easy fix.
Open System Preferences from the ? Apple menu and choose ?Mission Control?
Check the box next to ?Displays have separate space?
Log out or back in, or reboot the Mac for change to take effect
Once the Mac has booted back up again, you can place a window into Split View by holding down the green button or with Mission Control as described here, it should work without incident at this point.
The video below demonstrates this method of entering Split View:
Since this isn?t labeled as having to do with Split View it?s possible that it will change in a future version of OS X, but for the time being it absolutely works if you have that feature checked on. In a similar manner, showing the Dock on external screens also requires this checkbox to be enabled, whereas many Mac users may have turned it off in order to hide the menu bar from an external display or remedy high WindowServer CPU usage in OS X.
A big thanks to Pierre who left this solution in the comments, it has been confirmed to work both ways to both allow Split View, and disallow Split View if it?s unchecked.