TL;DR: Living fossils are the children of long-dead fossilised creatures that were good at being alike, other animals are the children that were good at being different.
The boring answer is that living fossils are around because nothing has happened to make them go extinct.
The more exciting answer is a tale of intrigue and lies stretching back hundreds of millions of years:
The lie that is the 'species'
The notion of a species is, of course, a lie. It's a useful lie, and one without which the equally useful field of taxonomy would be impossible, but a lie all the same. Every living thing is ultimately a unique individual unto itself. We group these individuals up and call them things, according to things that are more or less arbitrary, and we arrange these groups into bigger groups and fill books upon books with this method, because it makes it super easy to get a good overview of how life developed.
However, if we want to understand the details, we have to see past species, and look at the lineages.
The shark, our ancestor1
One such lineage that is of interest is that of the shark. Sharks are ancient: They are roughly as old as plants, and older than almost all land fauna. They are vicious, unrelenting killing machines, and all but perfectly suited for being this, with their sleek shape, their conveyor belts of teeth, and rough skin. At some point way back when, however, the lineage of sharks split in two: One branch that leads to fish, amphibians and eventually dinosaurs, squirrels and dolphins, and one that leads to sharks. The forebear to all these things were the early cartilaginous fish of the Cambrian ocean.
To answer the question as posed, we have to compare the success factors of the two lineages.
Why sharks are still around as living fossils
The real answer here is that, in a very real sense, they aren't. Modern sharks are no more alike their long-dead forebears than squirrels are. However, the biological niche inhabited by sharks (apex predator) is a successful one, and sharks, already existing in this niche, have had successful children, made more successful by being superficially and behaviourally alike to their parents, and so, the modern shark presents as a living fossil.
Why shark contemporaries are not still around as living fossils
The real answer here is that, in a very real sense, they are. Morphological differences aside, squirrels are no more different from their long-dead forebears than sharks. Rather, crowded niches have made generation upon generation of increasingly squirrely animals more successful by being superficially and behaviourally different from their parents, and so, modern squirrels have arisen.
- The shark is not in fact our ancestor, rather Cambrian early jawed fish are. The image of a shark is however far more evocative and easier to talk about.