They did not reverse time, they reversed the "arrow of time", meaning that time continued forward but entropy decreased a little, for a moment. Small temporary violations of the second law happens spontaneously all the time on a microscopic scale, wherever the thermal energy comes together in just the right way to be absorbed into an atom or molecule. It is the same thing as "wind assembles fragments back into unbroken object" except that the macroscopic version is so unlikely as to never actually happen.
In this case a quantum computer performed an entropy-decreasing operation. Basically they simulated one of those unlikely entropy-decreasing fluctuations, and because quantum computers utilize coherent quantum states, the simulation itself involved a decrease in entropy. But it was not a fluctuation in the quantum computer, the quantum computer was steered by careful control along the reverse path.
This method will not be used to raise the dead, unspill cups of coffee, take back stupid actions, or any of the other practical applications of reversing the arrow of time, because it can only be applied to quantum systems that were completely under external observation and control from the beginning.
EDIT: This answer was correctly criticized for relying too much on the description of the paper in the press release, which e.g. (in the section called "Reversing time on demand") does literally say that the IBM qubits were evolved into an "ever more complex" state, then subjected to the conjugation operator so that they would subsequently evolve in the opposite direction.
It is apparent now that the arrow of time which is the subject of this paper is not primarily the thermodynamic arrow of time, but rather a kind of quantum "orientation in time" (my phrase, not theirs) that is supposed to be more fundamental. The time-reversal operation creates an oppositely time-oriented state, and this has the subsidiary consequence that, if the initial state is one in which entropy is increasing, the resulting state will be one in which entropy is decreasing. That is, reversing the quantum arrow of time is capable of reversing the thermodynamic arrow of time; but it doesn't necessarily do so, e.g. if one is already in a maximum-entropy state.
Emilio Pisanty says that they were dealing throughout with pure states and these are by definition zero-entropy states, so the thermodynamic arrow never appears in the concrete systems they considered. Well, there are several definitions for entropy in quantum states, as demonstrated in these Physics.SE answers, but it's certainly true that none of those definitions appear in this paper. I definitely concede that this paper is not about the thermodynamic arrow of time. But do the processes they study nonetheless have a thermodynamic aspect, for some definition of entropy?
First, they study the theoretical likelihood of a wave packet spreading in the vacuum being time-reversed by spontaneous vacuum fluctuations. They argue that this is exponentially unlikely, in proportion to the scale of the spreading that is to be reversed. Then, for the actual experiment, they simulate a kind of elementary quantum scattering process, using two or three qubits.
As mentioned above, they don't talk about entropy in the body of the paper at all. Instead they introduce "time-reversal complexity". Their point here is that the more degrees of freedom or Hilbert space dimensions there are, the greater the complexity of the time-reversal operator, in the sense that you need more of the "parts" (serendipitous localized fluctuations or quantum logic gates) that it is made from.
However, it seems likely to me that the processes under discussion do actually involve an increase in "quantum entropy", for the right definition of entropy, and that this is related to the high complexity of the time-reversal operator. The wave packet as it spreads should experience an increase in "quantum Boltzmann entropy" (see the second answer at the Physics.SE discussion I linked above), and this is correlated with the fact that a time-reversing fluctuation becomes exponentially more complicated and therefore exponentially more unlikely as it continues to spread.
As for the simulated scattering process, the very fact that the "scattered" qubit goes from unentangled to entangled means that its von Neumann entropy increases. So while the overall multiqubit system is in a pure state throughout, the von Neumann entropy of the parts does increase and then decrease.
So to sum up: the experiment was fundamentally about reversing the quantum arrow of time, not the thermodynamic arrow of time. You could argue that what they simulated (a quantum scattering event) is one of the fundamental processes which, when occurring in vast numbers, gives rise to the thermodynamic arrow of time. However, since they only simulated one such event and not a vast concatenation of them, it's probably not reasonable to say that they reversed the thermodynamic arrow of time. Instead, all they did was reverse a one-time increase in the von Neumann entropy.
This answer should still be regarded as tentative, I am just answering this hastily and semi-intuitively rather than doing the due diligence of calculation. But it's been a week so some kind of correction seems better than none.