Translation:He wants to eat only potatoes for lunch.
Картошка is an instrument of his обед. =) Обедать is not an action of eating, so we don't use Accusative here like кушать картошку. The literal translation would be "he wants to have lunch with/by (?) potatoes" and not "to have lunch potatoes", maybe this way it would be easier for you to remember.
From my understanding, the reason there is no "C" here is because the phrase doesn't refer to two things combined (c) e.g. "рис с молоком" but to an accessory/tool/instrument to another thing. The second ones still imply "with" but there is no need to write the preposition (c). Just the person, the verb and the instrumental case.