The lystrosaurus itself was a rather mundane beast.  Initially, it was a lumbering low slung dinosaur about the size of a pig.  It walked with a "semi-sprawling gait," like an alligator.  It had a beak, and big plates for teeth, and was herbivorous.  It used its beak to nip off plants, which it ground between its plate teeth in a thoroughly pedestrian manner.

Lystrosaurus, in other words, was nothing special.

Until all the dinosaurs were wiped out, and it was the only thing left alive.

Now that's an exaggeration, of course, but not by much.  At the end of the Permian age, a huge meteor wiped out almost all life on Earth.  Everything died except for some insects, a handful of tiny rat-like proto-mammals, and the lystrosaurus.

In the millions of years to come, the lystrosaurus would comport itself well, adapting to every niche on Earth like Darwin's finches run amok.  At one point in the early Triassic it was the single most common vertebrate on Earth, comprising up to 95% of the animals in many areas.

Why did lystrosaurus survive and thrive?  This is one of paleontology's greatest mysteries.  We can't know for certain, but there are several theories.  One is that, as a burrowing animal, lystrosaurus was more accustomed to the stale air of underground burrows, and was therefore better able to weather the extremely high CO2 levels of the post-Permian Extinction atmosphere.

There are other physical and lifestyle characteristics of lystrosaurus that could have made it easier for it to survive.  But the problem is that all of these were shared by plenty of other dinosaurs - all of whom promptly went extinct.  For example, lystrosaurus was probably semi-aquatic, which might have helped it survive the catastrophe as well.  But there were a lot of semi-aquatic dinosaurs at the time, and they all died.

The latest theory is that lystrosaurus possessed an innate genetic flexibility, and an adaptability of nature which made it easier for it to survive.  By the end of the Permian era, most animals on the planet had evolved into highly specialized forms.  The problem with being a specialist is that if the situation changes, you're screwed.  (Just ask the Giant panda, which has to commute from its mountaintop retreat to the bamboo forests where its sole source of food grows.)

At any rate, lystrosaurus thrived in the new empty planet, devoid of competition and of predators.  It had the whole place to itself, and it got right down to business.  None of the mammalian predators at the time were even remotely big enough to pose a threat. And even though a lot of plants died during the Permian extinction, they bounced back rather quickly, without anyone left to eat them.

Lystrosaurus was a goofy and unattractive looking dinosaur, but it happened to be the right guy in the right place at the right time.  A certain amount of luck was definitely involved in lystrosaurus' takeover of the planet, even though its reign was relatively short-lived.