This is how I installed it on my machine (ubuntu):
php 7:
sudo apt-get install php7.0-zip
php 5:
sudo apt-get install php5-zip
Edit:
Make sure to restart your server afterwards.
sudo /etc/init.d/apache2 restart or sudo service nginx restart
PS: If you are using centOS, please check above cweiske‘s answer
But if you are using a Debian derivated OS, this solution should help you installing php zip extension.