It's a sentence fragment, so we could make it a complete thought and say "The flying bluebird found a moth." Now you can see flying is describing the bluebird. I think it's called a verb phrase in English, so you could also say "The bluebird that's flying found a moth." In Korean you only write that one way, which is the verb phrase coming before the noun. 날다 -> drop the ㄹ, add 는 for present tense -> 나는 파랑새가 나방을 찾았어요.
But it's not even a full sentence. It isn't uncommon for someone to speak like that in very casual situations. Example, a girl says "난 bambi" to sound chic or w/e, and drops the copula altogether (I've heard this before) when introducing herself on a youtube video.
So this is a really mean thing to translate without context as just a sentence fragment.