Specimens with the remains of ingested animals are sometimes found, but I've never heard of this kind of "Russian doll" fossil before. Truly remarkable.
And now that I've finished marveling at it, it's time for me to return to character and bemoan the nomenclatural inaccuracies!
First, what are the actual animals involved? It's an Acanthodes bronni, eaten by a temnospondyl, eaten by a Triodus sessili. Is the headline accurate? Let's establish a few ground rules first.
Most taxa (at least of those that have been around a while) are based on living organisms. When related fossil organisms are found, there is often some dispute as to whether to include them or not. For example, when Archaeopteryx was discovered, people couldn't decide if it was a bird or not. It had feathered wings and bird-like feet, but also had teeth and a long tail. Did it fly? Who knows.
In the past, some people have recommended letting taxa extend as far as possible, encompassing what is known as the total group. A total group includes everything sharing closer ancestry with the living members of a group than with any other living organisms. This approach can run into some serious problems, though. For example, it would make all dinosaurs birds. More generally, it lumps the earliest, barely differentiated members of a lineage in with their derived, living descendants. This encourages the use of unjustified inferences. For example, I know that all living birds have tertial feathers (long feathers along the upper arm), so, since Archaeopteryx is a bird, it should have them, too, right? WRONG. Analysis of the available fossils has yet to indicate the presence of any tertials.
This is why many others encourage the use of crown groups for common taxonomic names. A crown group is the final common ancestor of certain living organisms, and all descendants of that ancestor. Using a crown group definition for Aves, for example, would limit it to the clade of modern birds, thereby excluding Archaeopteryx. This practice has the effect of discouraging unjustified inferences.
One other term to note here is stem group. This is simply a total group minus its included crown group. For example, a stem-avian is anything sharing closer ancestry with birds than with other living organisms (e.g., crocodylians), but outside the crown group Aves. Archaeopteryx and all the classic dinosaurs are stem-avians, as are (probably) pterosaurs.
Cladograms (somewhat abridged) showing the avian stem group.
Is Acanthodes a fish? Well, yes, they got that right. "Fish" is a really, really broad category, tantamount to "any craniate that isn't a tetrapod" (more or less). Specifically it is an acanthodian, which makes it a stem-osteichthyan (Osteichthyes being the crown group that includes bony vertebrates, e.g., ray-finned fishes, lungfishes, coelacanths, and tetrapods).
Are temnospondyls amphibians? O.K., this is harder. Traditionally, Amphibia was used as a "wastebasket taxon" for any tetrapod that was not an amniote. More recently, it has been limited to one of two groups: the total group including frogs, salamanders, and caecilians; or the crown group including frogs, salamanders, and caecilians. There is actually quite a bit of debate on this matter and I don't expect it to be resolved any time soon. According to the crown group usage, temnospondyls are definitely not amphibians. But according to the total group usage, they might be. (But they might also be stem-tetrapods or stem-amniotes.)
Is Triodus a shark? "Shark" is a really abused term in paleontology. (In fact, acanthodians are sometimes called "spiny sharks" even though they're not even chondrichthyans.) Among modern animals, the term is generally limited to Selachii, the flatter members of which are called skates or rays. Triodus is a stem-selachian, closer to sharks, rays, and skates than to ratfishes (Holocephali). So is it a shark? If we want to limit unjustified inferences, then we should say no.
Cladogram of the taxa in question, with related extant taxa.
Okay, smarty, how would YOU dumb this down? Argh, that's pretty hard. Personally I think the lay public might be ready for the term "stem-", but I seem to have a bad habit of overestimating the lay public. Anyway, the best I can come up with is "Stem-Shark Ate Limbed Creature Ate Fish." Doesn't roll off the tongue, you say? Well, I guess this is why I'm not a journalist....
(References to be added later, if I have time.)