Clicking 'unpublish' puts a 'sitter found' message at the top of the ad and removes it from the list of available sitting positions so it's no longer in the search results. It also removes the contact message box from the bottom (so the home owner can no longer be contacted from their ad page).
The ad isn't totally removed for a number of reasons the main one being that sitters still have notification emails in their mailboxes with links to the page. By keeping the page and adding the sitter found message, if they click the link to it, they will see that the position is no longer available.
If there is something on the page the home owner no longer want's anyone to see they can always change the text on the page.