Option 1 - Release version
Before downloading the entire repository using the 'Clone or download' button, make sure there is no actual release (In most cases there is, to keep development organized):

If so, click on Releases Tab and download the latest release (zip
or single py
file):

Option 2 - init.py folder (Clone or download)
There are lazy authors so in some rare cases the 'Clone or download' is the only option. Search for __init__.py
in the repository and copy the folder containing this file to your Add-on directory:
- Linux:
$HOME/.config/blender/<version>/
- OSX:
/Library/Application Support/Blender/<version>/
- Windows:
%USERPROFILE%\AppData\Roaming\Blender Foundation\Blender\<version>\
You can get also get the folder per machine using the python console: bpy.utils.script_path_user()
and / or bpy.utils.script_paths("addons")
Further reading: Addons: Only some .zips will install