First of all, let's get some important 'sociological' aspects out of the way:
- While website you've linked to, phys.org, tries to pass itself off as a science-journalism site, it is nothing of the sort.
- Instead, its core business is to aggregate press releases written by universities themselves.
- For most of the press releases they publish, phys.org does not do any vetting at all of the contents of the documents, nor do they do any independent journalism or check with independent experts.
This means that the text you've linked to was written by someone (a university press office) with a direct financial stake in the impact of the piece, and it was not verified by anyone, and neither the writer nor the editor checked with any independent experts to confirm what they were publishing. That's fine if it's presented like what it is (i.e. promotional material), but it is unethical to present it as science journalism (which it is not). If it sounds like a "newspaper" taking claims about steel production in Russia from the Soviet Union's government and running them unchecked, that's because there's no difference between the two.
The same is true, by the way, with EurekAlert, and with Science Alert, whose takes on the subject are here and here. Notice any similarities?
Moreover, a word about the journal. The paper in question was published in Scientific Reports, whose refereeing process only checks for correctness, and not for interest or impact. (And, frankly, I wouldn't say that it has a great reputation for its correctness checks.) This doesn't impact the paper itself, but it does bear keeping in mind.
Anyways, on to the paper.
Suppose I throw an elastic ball at a wall, such that it hits the wall horizontally and bounces back:
Assuming that the ball is completely elastic, then the collision with the wall will reverse its velocity, and it will follow exactly the same arc back to my hand. Why is that? Basically, because newtonian mechanics doesn't have an arrow of time - its laws of motion are completely reversible, which means that if you keep all of the particles' positions intact, but you reverse all of their velocities, the system will track back on exactly the same trajectory it came from, only backwards.
The same is true within quantum mechanics: the laws of microscopic motion are completely time-reversible, though the operation of "reverse all of the velocities" is somewhat more complicated. Technically, you need to take the wavefunction $\psi(x)$ and replace it with its complex conjugate, $\psi(x)^*$, and this is what Lesovik and his coworkers have done: they've devised a way to take a wavefunction and reverse its complex phase. And once you do that, the system will follow back on its previous track, exactly like the bouncing ball in classical mechanics.
Image source: Sci. Rep 9, 4396 (2019)
So what does the current paper have to do with the arrow of time? Nothing at all, except for hype. The arrow of time emerges as a concept of statistical mechanics, in which the complexity of the processes in a large system is such that, although the individual microscopic laws of motion are reversible, the large-scale dynamics are not, since it is too difficult and unlikely to fully time-reverse the state of the system at any one time. This is primarily framed within classical mechanics, but there are equivalent versions within quantum mechanics, once you introduce the concepts of mixed states and thermodynamic ensembles.
However, the current paper does nothing of the sort. They work with pure states and with fully coherent dynamics (instead of mixed states and partially-coherent dynamics, which would be required to deal with thermodynamic concepts or the arrow of time), and this makes them unable to address any issues with entropy increasing or decreasing, or any of the interesting stuff on the topic. The authors talk a big game about entropy and surrounding topics in the introduction, but that's where it stops - they do not measure any entropies, so they're completely unable to say anything meaningful about "reversing the arrow of time".
So, let's have a final run through your specific questions:
researchers have reversed time in a quantum computer
No, they haven't. They reversed the direction of travel of a quantum evolution and watched it travel back, exactly like a classical elastic ball bouncing from a wall. There is some technical merit in the practical implementation, but nothing more.
and violated the second law of thermodynamics.
They did nothing of the sort. The paper effectively works at zero temperature and with pure states, so the entropy is zero throughout.
What does that mean for physics?
Exactly the same as a classical ball bouncing from a wall.
Will it allow time travel?
Not at all.