Stop WordPress from sending any emails. ANY!
You probably have another plugin that adds its own implementation of the
wp_mail() function. Try disabling some plugins.
You probably have a plugin that is sending emails via some other method, like directly using the PHP
mail() function, or directly implementing an SMTP client. Not much I can do about that...
The plugin replaces the standard WordPress
wp_mail() function with a function that sends no emails. Nada. Zip. Silence.
Behind the scenes, it creates a private copy of PHPMailer and allows the system to interact with it, but silently suppresses the functions that send emails. The standard WordPress filter and action hooks are supported, so plugins that register hooks for those will still function as normal. It just doesn't actually send any emails.