Most likely they're referring to the C# part being written for Microsoft's "ASP.Net Web API" framework, rather than just being a web API in general.
That framework is designed for C# programmers and most of the resources (tutorials/stackoverflow answers/etc) around it will assume you are writing C#. So it makes some sense to use C# for that part of your application even if most of the underlying business logic is implemented in separate F# libraries.
That's probably why they specified it - to explain why they're using both C# and F#. I can see how it could be confusing, especially if you're not familiar with that rather generically named framework.
