I un-zip by "extract all" and navigate to unzipped files but the source.gdb doesn't open in the ArcMap window
In your picture you try to add new Connect To Folder.
Connect To Folder will add link to browse window " Add Data" to go directly to this folder.
the solution is:
Choose in Connect To Folder "ElectoralGIS" folder (as your picture). And you will find that folder in Add Data window in the menu which is next to "look in : " .
then add .gdb files
Retrieving data ...