I've only played one single game with this feature: Chrono Trigger.
The feature simply lets you restart a new game with the data from a previously saved file, allowing you to kick ass with the Rainbow from the very start of the game.
This feature also eliminates "miss-able items", because you can always reclaim anything on your next playthrough, or the next, or the next.

New Game+ rocks, so why do so few RPG:s use the device?