Nice error message Microsoft. Why not just say, "Screw off. We don't care about your old projects." This is why I avoid them whenever possible.
However, in this case, it wasn't possible. It was some reference code for the protocol of a new thing I'm working on. It was done in Visual Studio 2013. I'm using Visual Studio 2015.
After doing a bunch of stuff that was totally useless (I am not very strong in .NET), I happened up on this simple solution...
Open the .csproj file in a text editor.
Change
12
to
14
Done.
Or, if you need more detail:
Change
<Project ToolsVersion="12.0" DefaultTargets="Build" xmlns="http://schemas.microsoft.com/developer/msbuild/2003">
to
<Project ToolsVersion="14.0" DefaultTargets="Build" xmlns="http://schemas.microsoft.com/developer/msbuild/2003">
After that it opened with no problem and I was able to build and use the project.
I have no idea what the ramifications of this could be in the long run. If I find any that are adverse, I will write it down here.