The words died, dyed sound the same but have different meanings and spellings. Why do died, dyed sound the same even though they are completely different words?
The answer is simple: died, dyed are homophones of the English language.
Simple past tense and past participle of die.
Coloured or tinted with dye.
Simple past tense and past participle of dye.
Definitions from Wiktionary, Creative Commons Attribution/Share-Alike License 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").