This is likely a Visual Studio problem, not just an Atmel Studio problem.
Let's say it's the way VS works. There is no real problem here with VS, and also Eclipse is very similar.
The real issue is that the video shows only how to add include paths, as the "library" the YouTuber was using had to be included as a .c file; why would one want to do that is beyond me, in the general case, but maybe it's an Arduinish thing.
If that were the case, everything would have worked.
The library Simon is using is provided instead as a set of include (.h) and source (.c) files, as most sane reusable C code is.
There are clear usage instructions on the library site.
In some cases the library can be pre-compiled (to a .a or .lib file, using a separate project), the includes placed in the common include directory and the library added to the Libs folder, since I see a setting.h file, I would expect this is not an option.
What's impossible to do is linking a .c as a compiled library: probably something has been messed up in the various attempts made, hence my suggestion to start over.