If you have root access to your server, you may be able to see the system-level mail log. The location will vary from system to system, but /var/log/maillog is usually a good guess. It will have the most information for debugging mail sent directly through PHP.
One of the challenges of SMTP is that the initial connection is off of your server, so access to logs to help with debugging is very limited.
Do you get the same "Could not authenticate" message when you change the port from 465 to 587?
To debug the SMTP connection, you will need to edit the SMTPMailer.inc.php to add more logging to your PHP error log to discover the source of the SMTP authentication failure. You will need to be comfortable with PHP to do this. We can offer guidance, but can't give you step-by-step instructions.