First, to answer your question as to who utters the sentence "What brung ye up this air way off the branch," from context it is "she."
The narrator who speaks in the first person always speaks standard English. It is the other character "she" who speaks some type of pidgin English, so there is no doubt in my mind "she" is the person who says that sentence.
For your second question, I do not have enough information except to guess. Again, the second person is not speaking standard English, so I do not know why she uses "air" or "branch." After all, she says non-standard words such as "who *mought*" and "what *brung*"
However, my best guess is that she asked "what brought you all the way here off the fork (in the road)." (i.e., "what brought you off the beaten path to come all the way here?") Alternatively, the narrator visited the "branch" of some office, company, or building, but again I do not have enough context to say for sure. You will have to see where the narrator visited "her" in this story to have a better sense of what "branch" means in this context.