Why do the proportion of predators increase at mass extinction events?
The data used in the graph you posted seems to be from Bambach. 2002. Supporting predators: changes in the global ecosystem inferred from changes in predator diversity (in Kowalewski &amp; Kelley. The Fossil Record of Predation) - see fig 19b - and, if so, is specifically dealing with marine predators.

I don't have time to look at that paper closely (it is quite long), and paleoecology is not my field, but a couple of relevant sections are:


  However, the diversity of predators and the proportion
  of total faunal diversity composed of predators have both increased over time, implying that ecosystems have
  increased their ability to support either more predators or more specialization among predators. Also, turnover
  in diversity dominance among predator groups, with more energetic predator taxa replacing or being added to a
  fauna of less energetic groups, implies that the energy available in marine food webs has increased. 


and


  If
  the efficiency of energy transfer between levels in
  the food pyramid has stayed nearly constant (and
  there is no evidence that it has not), then a reasonable
  hypothesis for the increased proportional diversity
  of predators is that either the biomass of prey species
  has increased or the populations of prey taxa have
  become more stable (or both), permitting greater
  specialization by predators on more reliable food
  supplies. Thus, predators could increase their
  proportional share of total diversity by specializing
  on fewer prey taxa.


So according to the author, energetics, energy flow and predator specialization seems like likely reasons for the change. However, the paper also discuss factors such as rates of fossilization and also the fact that predators are generally rarer than prey means that there is probably more stochasticity in the fossil occurence process, which could affect estimated turnover rates and maybe also the proportion between predator and prey. You'll have to look at the entire paper for a fuller picture, and maybe somebody more knowledgeable can fill in later. I also think that you need to look at radiation patterns in certain taxa to fully understand these stepwise changes, and also to consider oxygenetion events as possible underlying reasons, but these are just my speculations.

The reasons given above also in one sense echoes the "biotic" explanation for the latitudunal diversity gradient (see e.g. Gaston, 2000 and Mittelbach et al, 2007), where high levels of primary production can support more specialized predators/parasites in more complex food webs, then leading to higher species diversity. So there might be interesting connections to contemporary patterns and processes of biodiversity as well. However, I haven't seen a compilation of %predators over different biomes or latitudes to match the graph in the question. Also, the "biotic" explanation for the latitudunal diversity gradient has been criticized and is not without problems (see papers above for more).