if the project is simple, simply running the command make in the folder with the make file should work.
if the program is complex, you may need to edit the makefile or do other things which should be carefully explained in a readme file.
note that make is a unix utility, and if you are trying to build this on windows, there is likely going to be more effort involved. You can try using cygwin, which is a windows tool that gives you unix commands (including make and g++) and can build most programs on windows directly.
you can also try just dumping the source into an IDE and seeing if you can beat it into submissions, but some makefiles have project settings all over the place.
Sometimes importing an existing project from GitHub or other source is just a nightmare because of a big amount of specificities according to the original source. Sometimes it works well because you have all dependencies inside the main project which has been created with the right properties, but it is rather rare. Often the author has created his project with properties which are typically from his own station (folders and subfolders). Very often, it depends on the skills of the developers. It seems to me that they want to share their projects, but not too much :)
and.... life is too short to fix a train wreck. Unless it is exceedingly one of a kind and extra special, throw it away and find another one. Eventually you find one that works without 2 months of trying to fix it. If its of any use at all, it will either have a working visual studio project, or a working makefile, straight up without any tampering or downloading 500 other things.