This could be a seriously dumb question, but I have checked in the terminal that my computer is using swift 1.2 right now:
$ xcrun swift -version
Apple Swift version 1.2 (swiftlang-602.0.53.1 clang-602.0.53)
The release version of Xcode (6.4) only supports Swift 1.2. You need to get the latest beta of Xcode 7 (beta 6). Once you have installed that (and you use it instead of the 6.x version) it will automatically use Swift 2.0.
You may also need to run
xcode-select from the command line to switch the command line tools to use Swift 2.
xcode-select --switch /path/to/Xcode7/beta
You might need to run it with