Go to release 3.15.1 packages page and download carrot2-csharp-api-3.15.1.zip.
After download completes:
Extract the archive to some local directory
Look in the examples directory for code examples
See Carrot2 Manual for instructions on how
to compile a C# with Carrot2.
Note: Carrot2 C# API requires .NET Framework version 3.5 or later.