The zip contains a subdirectory. Only the subdirectory should be uploaded to your site plugin folder. If you want to upload a .zip file in the browser, only the subdirectory should be zipped first.
Thanks--zipping the subdirectory, then uploading it, worked to install the plug-in. (If it's not too much trouble for you, putting it in the WordPress Plugin Directory, http://wordpress.org/plugins/, would let WordPress users find it easily.) Unfortunately, when I tried to activate it I got this error:
Parse error: syntax error, unexpected T_FUNCTION in /home/content/61/6806261/html/wp-content/plugins/thedaywefightback/thedaywefightback.php on line 13
The official site has chosen a similar plugin that has some additional options you can configure. They've also added good instructions on how to install it.
https://github.com/modemlooper/thedaywefightback.wp