This island is home to the Legendary Pokémon Darkrai. Currently the only ways to get this Pokémon are by trading for it with a friend or somehow get the event-item Member's Card
First you have to give the Lunar Wing to heal Eldrich's son to start the Darkrai event. After receiving Member's Card from the Nintendo event, go to Canalave town and go to the inn in the north of the town. Once you enter, a weird man will put you to sleep, and you'll find yourself on Newmoon Island.
Capturing Darkrai[edit]
Go through the entrance to the forest. You will see Darkrai hovering there. Walk up to it and challenge it. It is level 40. Like with all Legendary Pokémon, save before battling it. Start out with a Pokémon that can put it to sleep then switch to a Pokémon with False Swipe and get it to 1 HP. Once you get that the start throwing Poké Balls and putting it back to sleep if needed.
Once captured or defeated you will return to the inn and a small cutscene will happen with Sailor Eldrich.