The words aerie, aery, airy, eyrie sound the same but have different meanings and spellings. Why do aerie, aery, airy, eyrie sound the same even though they are completely different words?
The answer is simple: aerie, aery, airy, eyrie are homophones of the English language.
A house or stronghold perched on a height.
The nest of a bird, such as an eagle, built on a cliff or other high place.
Ethereal.
Variant of aerie.
Of, relating to, or having the constitution of air.
High in the air; lofty.
Open to the air: airy chambers.
Performed in the air; aerial.
Variant of aerie.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").