Fullmoon Island is only accessible after getting the National Pokedex. You will find that Sailor Eldrich's son is stuck in a nightmare, muttering about "...Dark...". This might mean Darkrai has done something to him. Talk to the man outside the house, Sailor Elderich and he will take you to Fullmoon Island, where a Pokémon, Cresselia is said to reside. The Lunar Wing is the only thing that can heal his son. There is nothing on the island except a small glade where Cresselia is sitting in a pond. Talk to it to be shown a picture of it. Afterwards Cresselia disappears (like Mesprit).
The Lunar Wing is where it once was, so pick it up and heal Elderich's son.
Note: The only reason to come back here is that one section of the Underground is inaccessible without it. You must not be in the glade where Cresslia is found, or else the Explorer Kit will not work.