If the project uses Visual Studio then the git repo should already contain the project files. Since you say you're getting an empty project in Visual Studio I'm guessing that the project doesn't use VS.
If the project itself doesn't use Visual Studio then Visual Studio's git integration won't be much help. It's just another way of downloading the source code, basically.
To use Visual Studio with such a project you'll have to do the same things as you would after downloading any source: create a solution and projects, add the files, convert whatever build system the project uses over to VS, etc. Once you've converted the project to VS then you can commit the files you've added and begin taking better advantage of VS's git integration.
Most of Visual Studio's features rely on Visual Studio having a deep understanding of the project that comes from Visual Studio being the project's build system. If you don't want to convert a project to use VS as the build system then what you'll get from VS is little more than a text editor. If that's all you want then you can simply open the source code files in VS and not bother with creating Visual Studio project files.