This is because the SampleVS2010.vcxproj is a C++ project, not a C# project.
For information on setting up a C# project see: https://developer.leapmotion.com/documentation/csharp/devguide/Project_Setup.html
The process should be the same or very similar to setting up any C# project that uses an external API. The biggest difference is that the Leap Motion C# library also loads a native, unmanaged library. This native library, LeapC.dll, must be copied to your executable folder. You can do this manually or setup a post-build copy step in Visual Studio.
Note that in SDK 3.1.3, the native dll to use for C# projects is LeapC.dll, not Leap.dll. The other programming languages still use Leap.dll, though.