Flesh is a synonym for meat. I don't like it when the fashion presenters describe see through sleeves as showing some flesh (meat.) A bit crude imo. Better to say that you can see skin through the material. More delicate. More lady-like.
Before butchers were butchers (French origin word) they were flesh mongers... with more a Germanic origin, I think). Meat used to be a more generic word for food rather than specifically about beef, pork, lamb etc.