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.
One difference in icon reordering between the Apple TV and other iOS devices is that the lack of a touch interface means that only one icon at a time can be selected and moved on the Apple TV. On other iOS devices, touching any icon for several seconds causes all icons to wiggle and multiple icons can be moved or deleted before exiting the mode. There is also no sign yet of an ability to delete or hide unused icons on the Apple TV menu.
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.