Composer is a package manager for PHP, it downloads packages from the packagist.com repository and places them in a nice structure along with an autoloader that would allow you to easily use them.
You can download composer as documented in its official website : https://getcomposer.org/download/
After doing so I usually place it in the path folder in order to be able to call it from anywhere, though that's just a preference.
Once you have Composer up and running, create a composer.json file in which you'll put the packages you'd like to use for your web application - in this case "giggsey/libphonenumber-for-php", without the quotes. The github link you posted contains the exact json structure :
{
"require": {
"giggsey/libphonenumber-for-php": "~6.0"
}
}
Finally, call composer.phar install on the console and it will take care of everything for you.
To instantiate the library, first include the vendor/autoload.php file then simply follow the instructions in the "Quick examples" chapter of the github link.
Note that everything I said about composer is detailed on the official documentation : https://getcomposer.org/doc/01-basic-usage.md