I haven't dug into the paper in too much depth, but I think that it is a little hard to give a precise definition of what they mean by this term in this paper, because it is used in the context of a complex phylogenetic statistical model that they analyze using MCMC simulation sampling or Ornstein-Uhlenbeck (OU) modeling.
For MCMC they appear to be estimating ancestral states (of REM sleep proportion for instance) along the nodes of the phylogeny based on the estimated values of the character for each tip (living organism) of the tree. This is probabilistic and they run the MCMC model until it converges at a reasonable set of parameter values for the overall phylogenetic model, and then they sample reconstructions of these ancestral states. You could view each of these reconstructed states as having its own "selection regime"; however they are not directly describing those regimes. Instead, they are noting only "shifts" in the selection regime, which they are defining as MCMC reconstructions that show a change on the human branch that is larger than that for the rest of the tree of primates. So, it's kind of a black box, because we only know the selection regime in terms of it being "different" between humans and the rest of the tree, as inferred by seeing a large shift in the state of the character (REM or whatever) in humans. That itself is a bit of a stretch of course, because large shifts can also happen without selection. But they're doing their best.
For the OU model, this model itself contains definitions of selection regimes, I recommend looking more into how OU models work to get more information about that. But according to my limited understanding, the OU process is trying to directly model changes in selection regime across the tree. So in looking at the black balls in Figure 2, those are direct OU estimates of the probability of such a shift. Again in this case, what exactly the selective regime is cannot be known from the paper, it is instead inferred to "shift" by the OU model based on changes in values of the character across the tree.