Its simply a way of saying some one has cheated but doing it in a rhyme. Eater rhymes with cheater. On a strict interpretation it is of course meaningless as eating has nothing to do with cheating. A similar phrase is liar, liar, pants on fire.

That is something children say to each other if they think another child is cheating during a game. Usually though it is said "cheater, cheater, pumpkin eater".