Marcher sur des oeufs, literally “to walk on eggs”, is the equivalent of the English expression “to walk on eggshells”. It means to be careful and sensitive in handling very sensitive matters, usually to avoid hurting or provoking someone. The expression appeared at the beginning of the 19th century. Obviously, it refers to the fragility of the eggshell and the idea that if you were to try and walk on whole eggs without breaking them, you would have to be very careful.
French with Valerie 