"I eat the fruit and the egg."
Translation:Ben meyveyi ve yumurtayı yerim.
ile and ve both mean "and." However, ile implies a much more closely associated relationship, two things you are treating like a set, and this often has grammatical consequences.
So these things are getting parsed in two different ways:
Ben [meyve ile yumurta]-yı yerim. = I eat the fruit and eggs--like you're choosing the fresh fruit and egg breakfast platter at Waffle Evi, or whenever you eat your eggs you eat them with fruit on the side. Since they are a unit, they only need one case ending at the end of the phrase.
Ben [meyve]-yi ve [yumurta]-yı yerim = I eat the fruit and the eggs. "Ve" can never treat things as close unit like "ile" can. It's like you're picking a couple things out from selections that don't necessarily have anything to do with each other. Since they are not a unit, they each take the accusative case ending.