This is a fascinating phenomenon – but fully within current theory. And there’s no “inversion of the arrow of time”, despite what the sensationalistic, misleading title seems to imply. From the recent paper (my emphasis):

Our results, over a range of pulse durations and optical depths, are consistent with the recent theoretical prediction that the mean atomic excitation time caused by a transmitted photon (as measured via the time integral of the observed phase shift) equals the group delay experienced by the light.

The theoretical explanation is given in this paper:

We examine this problem using the weak-value formalism and show that the time a transmitted photon spends as an atomic excitation is equal to the group delay, which can take on positive or negative values.

It is essentially related to the difference between phase and group velocity of waves.

One more example of how nature – as we currently understand it – offers amazing, fascinating, unexpected phenomena. It doesn’t need misleading sensationalism.