It might help you to think of a C++ header file as containing two major types of things: the class definition, which defines its member data and "interface" (not to be confused with a C# interface type) and "other stuff". The class definition part contains method prototypes and class member variables.
The good news concerning the prototypes is that you simply don't need them in C#. Clients of your class receive prototype information from the implementation itself via the assembly for the namespace. The data members are contained within your C# class implementation, typically as private members which are exposed through C# properties.
The "other stuff" mentioned above can include #defines, which you typically want to turn into const definitions in C#. Other things such as enumerations have equivalents in C# which you of course move into the .cs file for your class.
#includestatement is written. So in effect, even in C++, there is just a single source file for the translation unit, once the inclusion has been made.