How do I make WordPress send an email to the user when his account gets approved (manual by me)

What you are trying to do – approve a user in the admin and send an email – must be done using action and filter hooks, either by a plugin you’re using or your own code in your child theme or a plugin.

To approve a user, take a look at the user_register, edit_user_profile, edit_user_profile_update and load-user-edit.php hooks to get started. In your own approval code, you would then structure your email and send it with the wp_mail() function.

Try searching the net for detailed solutions, you should be able to find blog posts with ready code.