Fish don't fry in the kitchen conveys that things do not always happen where or in the way you expect them to.
This particular song speaks about moving up on the social ladder, becoming rich from poor. You expect the poor to live in poor areas, but the song shows that if you have opportunity and you persistently make the effort, this can change. Fish don't fry in the kitchen is a metaphorical images of the poor having access to success against all odds, so that you will find them, not in the poverty you would expect, but in a life of luxury because they moved up "to the East Side".
Having said that, we must keep in mind that it is a song after all, and although all the connotations I have mentioned are present, you cannot claim with the authority of a dictionary that this is the whole meaning of these idioms. Songs are very flexible, subjective and can be interpreted in many ways.
EDIT: I thank the downvoters for prompting me to check the information of the site which was accused of AI generated information, and to my shock, I could not find any support for that post. So I edited my post to delete it. It's a good lesson, next time I will check, as I never trust AI and would hate to propagate its gibberish.