This option which is visible to all privileged users beside each topic on the topic view (thread.php) allows them to move the topic to a different forum. Clicking on the link will open a pop-up window with a list of all the available forums.

If the user in question has permission to move the topic to the destination forum then the topic will be moved to their forum of choice once they click on that forum.

Otherwise beside the forum there will be a 'request a move' link. Clicking on this link will allow the moderator to request that the topic be moved to another forum, the form will allow the moderator to enter their reasoning behind their request. It will then be up to the administrator(s) or the moderator(s) of the destination forum to approve or decline the request via the Topic Exchange.