It is an installer. Look at the root of your drive. Do you have a Developer folder there?
Look n the SDK folder. Oddly enough, I'm running 10.8.3 but only have the SDKs up to 10.6.
And have (I think) downloaded the latest Xcode.
OK, from the Xcode faq...
To run the latest version of Xcode, make sure you are launching Xcode.app from the Applications folder. Xcode 4.3 and later is packaged on the Mac App Store as a single application and installs in your Applications folder as Xcode.app.
Previous versions of Xcode placed the tools and frameworks in the Developer folder. You can delete the older version of Xcode by dragging the Developer folder to the trash.
So don't bother looking in the Developer folder.
EDIT: The SDKs are in the app itself. Do a 'Show Package Contents' on Xcode and look for it.