This is set on the Emails section of the users preferences page. The default setting is to send an email to a user if someone quotes them, replies to their post, mentions their username, or invites them to a topic when the user is away from the forum:
If you are finding that users are not being sent emails for this, have a look at your default email level site setting. The default value of that setting is “only when away.” Make sure that it isn’t set to “never.” You can adjust the email settings of all users on your site by making changes to your default email level site setting and selecting the option to apply the changes historically.
I am not seeing any settings in your screenshot that would prevent emails from being sent when a user is mentioned. You can confirm whether or not emails are being sent when users are mentioned by going to your site’s Admin / Emails / Sent page and entering mentioned into the Email Type filter input. The screenshot below is from my test site:
I see you have the disable mailing list mode box checked (Disallow users from enabling mailing list mode (prevents any mailing list emails from being sent.)
Try unchecking that box and see if the emails get sent.
You will need to check the specific user’s preference because changing the global doesn’t necessarily change it for all users – historically that only changes it for new users who signed up after you changed the default.