The Chromecast is slowly worming its way into every part of the Google ecosystem, but there is at least one aspect you weren't supposed to see yet. There is incomplete screen casting support built into KitKat, but it has been surfacing in very odd ways for the last few months. Rest assured, you are not alone in spotting it. Update 7/9/14: It's live now
We tend to get a tip about this almost daily, but there's very little to report until Google actually implements the functionality. We appreciate every tip we get (seriously), thought we don't have much more information than you on this one. The Cast Screen tile seems to pop up for users of Nexus and newer Motorola devices when certain Chromecast apps have been used, for example AllCast. It's triggering something in Android related to an incomplete casting feature – that's all we can say for sure.
No one has, to our knowledge, gotten this native screen casting feature to work on a stock device – it just crashes when trying to connect to a Chromecast. It may also plug into Miracast right now, but there is something more going on. There are some third-party apps that aim to do the same thing, but having the built-in ability would be ideal. Presumably Google will fix this and expose it to users eventually, but right now all you're seeing is a stub of a future feature.
[Thanks to the many people who sent this in, and Scott for the screen]