What does 이 mean here?
이 means "this", as determiner modifying a noun/pronoun in most cases, but also as a pronoun sometimes.
Since 내(I) is modified by 이(this), 이 내가 날게 해줘서 is literally "For making this me fly" or "... such me fly".
"this me" or "this I" would be unacceptable in English. In Korean it is a bit less so, but not that much.
What is more common would be 이런 내가, where 이런 means "like this", "this way", or "in this state/shape".
So the phrases are a little forced expressions.
고마워 내가 나이게 해줘서.....Thank you, for letting me be me.
이 내가 날게 해줘서...............For letting (such) me fly. (It defies proper translation)
이 내가 is rather weird even in Korean, which is more flexible than English in terms of restrictions imposed by grammar and logic. It's a borderline usage of 이 - I would not use such a phrase.