Fish don't fry in the kitchen conveys that things do not always happen where or in the way you expect them to.
A site called Instructional Aids for Speaking & Communicating in English has a post on exactly this matter. About Fish don't fry in the kitchen it says:
The phrase “Fish don’t fry in the kitchen” originates from a popular song written by American musician and civil rights activist, Joe C. Turner, in the 1940s. The saying reflects the struggles faced by many African American families, emphasizing the challenges of finding stability and success.
This song became a cultural touchstone, capturing the essence of perseverance. The kitchen is symbolic here, representing a space of cooking, nurturing, and sustaining life. The idea that fish don’t fry suggests that not all endeavors can thrive in every environment. This reflects how context matters when discussing aspirations and goals.
Now, 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".