Aerie means the nest of an eagle, or other bird, of prey which is constructed in a high place: “He was able to see the Eagle’s aerie on the side of the mountain.”
Airy, a derivative of air, means well-ventilated: “The house and all its rooms were large and airy.”
It also means to go about in jaunty or lively fashion: “The children hopped around in an airy way.” / “The band is famous for its airy tunes.”
It also refers to something ‘imaginary, fanciful, dreamy, unrealistic’. For example: “Nayana’s head was full of airy ideas.”
Last but not the least, eerie means ‘so strange as to inspire a feeling of fear; uneasy because of superstitious fear; ominous, ghostly, spooky’. For example: “Graveyards can give anyone an eerie feeling.”
Usage example that uses these 3 words: Seeing the aerie of the eagle in the airy forest gave me an eerie feeling.