An object in motion will remain in motion, until acted on by an external force.
The real interesting thing here is that something can move, but have no energy - potential or otherwise. Unlike things we are used to, if a time crystal train hit you you wouldn't feel a thing. (Well, not exactly, but it gets the point across.)
The actual interesting thing about this crystal is that the atoms will spontaneously start spinning as you remove energy from the system. Ie, in the ground state (the lowest energy state) the atoms are in motion. This is in contrast to all the physical systems we are familiar with where the ground state is motionless.
The reason it is called a 'time crystal' is that the ground state 'breaks symmetry' in time in the same way that a regular crystal 'breaks symmetry' in space - in the sense that if you translate the system in space, you do not usually get back its original state, unlike a gas. And, this symmetry breaking occurs as you cool the system. Crystals self-organize in a periodic grid as you cool them from the liquid state, just as this time crystal will set itself in periodic motion as you cool it.
One consequence of the symmetry breaking is that the atoms will have to move either clockwise, or counterclockwise, and will "randomly" choose one of the two.
I'm about half-way through this and it feels more and more like it was written by someone who would also ask why a permanent magnet can support a load against gravity without expending any energy - it's the same fundamental attribution error.
EDIT: In fact finishing it, I'm really confused as to what's new here. Giant external magnetic field, and stuff just coasting in a circle? Nothing about this feels implausible in a classical sense with a ring of charged ions.
If general relativity is correct, that system will generate gravity waves which will take away energy and the orbit will decay. This is a small effect, however, so we generally don't have to worry about it. E.g., Earth's orbit is decaying by about the width of a hydrogen atom every few hundred years.
Since 1) the Nobel-winning physicist, 2) the peer reviewed journal AND 3) the main critic of his theory agree that there is something new there, you probably ARE confused as to what they really meant.
I think an appeal to authority is appropriate here. Time and again I've seen simplistic comments in tech forums, putting down new scientific theories based on some misunderstanding of what new they claim, with the argument that "it's just like" some other older theory.
And all this criticism usually based on a 10000 miles high explanation of the idea in some popular science article. As if that is enough to capture all the nuances of the thing.
I may not be quite proficient in Physics to be able to say what's the case here (then again I doubt many if any at HN are at the level of the article's professor), but I've seen the same kind of responses many times in comments about other fields of study.
Here, the theory is that the system is already in its lowest-energy state, despite the fact that part of it is still in motion.