Those objects are sea slugs, also called sea cucumber or trepang. Holothurids, lacking a formal brain or central nervous system, and rather primitive.
And yes, they are edible. Quite tasty, in fact. Sea cucumbers are very nutritious, and considered tonifying for middle-aged and elderly folks, especially good for people with arthritis and high blood pressure. Of course like many odd or suggestive ingredients, they are also often thought to be excellent either for the reproductive organs or the 'delicate' tissues.
But their main appeal is that besides a pleasant gelatinous texture, they absorb flavourings and sauces nicely.