This multi-tiered area is where the sylphs of Cloud City reside. By defeating the sylphs in battle, they will grant you their Essence which can then be exchanged with Alice at the Sylph Exchange for a true sylph that will accompany you during your adventures.
On each floor, you will find level 50 sylphs wandering around. Defeat sixty of these to obtain sixty Essencer which can be traded to Alice's Sylph Exchange for one sylph seal.
If you are lucky, a sylph might drop an Uncommon or Rare Sylph Seal (of the same type of the sylph you're trying to fight). If you are defeated in the Sylph Atoll, you will be teleported back to the exit portal and gain no Essences.