나는 is both a pronoun referring to oneself and one of the conjugated forms of 날다 (to fly). This is because verbs whose last syllable (excluding -다) ends with ㄹ actually drop the ㄹ when conjugated. When you conjugate 날다, it drops the ㄹ, hence "나는" in the adjective form.
Yes. I have a question too, though: is this more often used in speech than "파란새?" If so, why? It seems weird ro me that the noun form would be used here, which seems rare to me for Korean adjectives. Ooo or is it just a noun - like bluebird?