The words stile, style sound the same but have different meanings and spellings. Why do stile, style sound the same even though they are completely different words?
The answer is simple: stile, style are homophones of the English language.
A set or series of steps for crossing a fence or wall.
A turnstile.
A vertical member of a panel or frame, as in a door or window sash.
A comfortable and elegant mode of existence: living in style.
A mode of living: the style of the very rich.
The way in which something is said, done, expressed, or performed: a style of speech and writing.
The combination of distinctive features of literary or artistic expression, execution, or performance characterizing a particular person, group, school, or era.
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").