Shift right-clicking will open the Checkpoint's interface. It is laid out just like a player's inventory; after death, the Checkpoint's contents will be moved into the newly spawned user's inventory.
Due to a bug, the Checkpoint will not pull any items away from its inventory after death; it can be used to dupe items.
The Checkpoint was added before Minecraft added the Bed. The Checkpoint was planned to be removed in version 2.0 of the mod, but it never was.