For example, the world commonly refers to Korea and Japan with these English language names. But Koreans refer to their nation in their own language as hangul (or something similar) and Japan says nihong (knee-hoe-ngg) in their native language. I think China has different names in Mandarin/Cantonese/etc. too.
My question is, why did these Asian countries ever feel the need to use a foreigner friendly nickname for their nation?
Pretty much every country does this in every language, and there are different stories behind all of them. Germany is a good example:
Germany (English)
Allemagne (French)
Todesco (Italian - "German")
Deutschland (German)
For the ones you asked:
Japan = Chinese pronunciation of the characters 日本 (with a Portuguese accent)
Korea = Chinese pronunciation of an ancient dynasty on the peninsula (高麗)
China = from the Qin dynasty through the pronunciation of various ancient Indian languages
Honestly there are way too many of these to go through, as they are different not only by country but for many languages for each country.