-다 breaks an action off midway and 남다 says what remains: 먹다 남은 음식 is food left over from a meal someone was part-way through, 팔다 남은 떡 the rice cakes left unsold at market. Almost every line here is 먹다 남은, and usually the point is that somebody else already had a go at it. Plain 남은 음식 is just 'remaining food', with no story about who started it.