If the the Admin panel ->Settings -> New Message is enabled then when you add a message to the ticket it sends an email.
The problem I think is that the text at the bottom of the ticket screen (Staff) is misleading. It says,
"Ticket Status: Close on Reply " (or "Ticket Status: Reopen on Reply ")
but it should really say "Ticket Status: Close on Post Reply " or even "Ticket Status: Close now on Post Reply."
because that check box is closing or re-opening the ticket when you hit the "Post Reply" button rather than meaning when the recipient "replies" to the email. You will find that the ticket status toggles immediately you check that box and hit the Post Reply button (i.e. was Open -> Closed)
So the workflow would be, fill in a message and then check that box and then click the Post Reply button and the ticket will close, the message would have been sent to the originator as long as the config setting Admin panel ->Settings -> New Message is enabled.
The user gets a default template message like,
-- do not edit below --
MY USER,
Our customer support team personnel has replied to your support request #123456
I rebooted the dohickey thing-gummy and removed the ratbait catcher.
We hope this response has sufficiently answered your questions. If not, please do not send another email. Instead, reply to this email or login to your account for a complete archive of all your support request and responses.
http://www.example.com/osticket/tickets.php?e=my.user@example.com&t=123456(http://www.example.com/osticket/tickets.php?e=my.user@example.com&t=123456)
If they check that ticket they will find it closed and then they can post a message and re-Open the ticket.