This pattern is revealed by the branches in these trees. If you think about each branch on the tree as an interval between two speciation events, then the lengths of those branches should be related to diversification rates; if diversification rates are high, speciation events will be closer together, and the branches in the trees will be shorter. The authors suggest that one explanation for this pattern is density-dependent cladogenesis. That is, the rate of diversification slows as species accumulate.
I like this paper because this particular tree shape, with short branches near the root, is one that I commonly observe in my own phylogenetic trees. If these results are general, we might be able to confirm something that evolutionary biologists have suspected: species interactions affect diversification rates. We postulated in 2003 (paper here) that such early bursts in diversification rate might be associated with bursts in the rate of morphological evolution, another characteristic pattern of adaptive radiation. This hypothesis has not yet been evaluated across a large enough number of trees to form any conclusions.
This paper reminds me of another general pattern in macroevolution: phylogenetic trees are more imbalanced than one would expect based on most null models (see Mooers and Heard 1997 & Blum and Francois 2006). Such regular patterns at macroevolutionary scales are hard to come by. When we find general patterns at this level, we have learned something deep about the process of evolution. Interestingly, there seems to be some relationship between imbalance and slowdowns in the Phillimore analysis; more imbalanced trees tend to show stronger slowdowns. Perhaps this is because the density dependence sometimes has a lineage-specific component.
Also, don't miss the nifty set of simulations in this paper showing that there is a slight bias in the test the authors are using, but that the pattern in the data is too strong to be explained by that bias.