What does the 'MageMail API Access Denied' error message mean?

This means that when the MageMail server attempted to query the API running in your Magento store, it got a 403 response code.  This usually means that there is a security rule running in your web server that's preventing it from querying the API.

You should check with your hosting company to see whether that's the case.   You can tell them the URL being requested is:

<a href="https://www.example.com/index.php/magemail/api/queryJsonSerialized/route/">https://www.example.com/index.php/magemail/api/queryJsonSerialized/route/</a>

And the IP address that this request is coming from is:  45.55.38.132.