The error message "Your PHP installation appears to be missing the MySQL extension which is required by WordPress" can appear when you are using a PHP version which is incompatible with your script's version (your WordPress installation is most likely outdated).
Since the MySQL extension is deprecated (no longer supported) from PHP 7.0 and beyond, changing your PHP version to 5.6 or lower will fix the issue.
This can be done via your website's control panel's PHP Configuration section.
NOTE: The best solution would be to update your WordPress installation. To find out exactly how you can do that, check out this tutorial :)