Based on EdgeCase's fantastic Ruby koans, the goal of the ASP.NET Web API koans is to teach you ASP.NET Web API through testing.
When you first run the koans, you'll be presented with a runtime error and a stack trace indicating where the error occured. Your goal is to make the error go away. As you fix each error, you should learn something about ASP.NET Web API.
As each ASP.NET Web API user will be using a different language, switch to the branch matching your language to find the koans relevant to your language. The README should reflect additional instructions.