The phrase is never actually explained in the show, but it's thought to be a play on traditional phrases like "By the Great Horned One" (referring to the devil). In the context of the Flintstones' prehistoric world, "horn spoon" likely refers to a large utensil made from the horn of a prehistoric animal.
While not a real phrase, "By the great horn spoon" has become a popular catchphrase and is often used in a humorous way.