The feature is accessed by holding down the center Select button on the remote for several seconds, after which the selected icon begins to wiggle as is seen on other iOS devices when in reorder/delete mode. The wiggling icon can then be moved by using the directional buttons on the remote, with other icons automatically adjusting to make room for the icon being moved.
On the current Apple TV software, holding the Select button for several seconds puts the device to sleep, and it is unclear whether this functionality is being discontinued or reassigned to a different remote shortcut.
Article Link: iOS 6 Beta Allows Reordering of Icons on Apple TV