"No, I'm okay."
Contractions are common in Korean.
The expression 괜찮습니다 is itself also a contraction from 空然하지아니하다! See: https://korean.stackexchange.com/questions/129/is-there-any-word-that-varies-in-length-when-written-in-hanja-or-hangul/2332#2332
Technically, this is not the way that a Korean in a typical conversation would say "No, I'm okay." That the way we would say it in English, but it's backwards for them (still kinda confusing to me..). So they would say "예, 괜찮습니다" which is sometimes translated to English as "No, I'm okay." It's weird and I can't explain it further, but after listening to the language for so long, I'm somewhat used to it..
That's because 네 and 아니요 don't really mean "yes" and "no" but more like "I agree" and "I disagree". Like, if someone asks "Didn't you do that?" and you did not the thing, you don't say 아니요, because then you would deny the sentece, you say 네, meaning "yes, you are right, I did not do that". Really confusing but gets easier once you catch up with the logic behind it.