(Please note, I'm not an expert on CC, but I play one on StackOverflow. Some of these statements may be slightly incorrect)
Code Contracts aren't just a collection of methods in an assembly you reference and call in order to perform validation. Code Contracts are a set of types and tooling that actually rewrites your IL after compilation to perform static and runtime checks. You can read more about this rewriter here.
They are still not an official part of the framework or of Visual Studio. You have to download and install the tooling for visual studio before it works.
Editable ASP.NET websites perform the compilation of aspx/cshtml files on first load. As Code Contracts require your compiled assemblies be rewritten, this would have to be included in the ASP.NET compilation stage. I'm not sure how to do that (I'd visit the forums for that info), but it definitely won't happen by magic.
The error suggests that compilation is not being performed correctly either in VS or by ASP.NET. Either way, the only way to fix it is to ditch CCs altogether, or do what the dialog says.
I'll note that Daniel Cazzulino has some CC-like code available via nuget that includes many helper methods that behave similar to Code Contracts. Things like Assert.NotNull(someArgument, "someArgument");
. You can find them if you search for his account.