I had heard previous explanations to this phenomenon that has been both observed in the fossil record and in real life. The main explanation, which is slightly simplistic I think but certainly perfectly passable, is that a relatively sudden change in the environment triggers relatively sudden evolutionary change for life to adapt to it and then, once all the new niches have been filled with species evolved to be fully adapted to each one, evolution slows down to its usual slow rate again. This makes perfect sense if you think about it very carefully!
Although this link below doesn't mention 'punctuated equilibrium', it obviously is a good study that helps to clarify why it happens and I personally find it quite interesting:
"...The rate at which new species evolve is limited by competition for ecological niches, report scientists from the University of Chicago in Nature on April 30. The study, which analyzes the evolutionary and genetic relationships between all 461 songbird species that live in the Himalayan mountains, suggests that as ecological niches within an environment are filled, the formation of new species slows or even stops.
"Despite the great diversity of environments and ability for species to move between areas, evolution in eastern Himalayas appears to have slowed to a basic halt," Price said. "Other species have formed elsewhere, such as in China and Siberia, but most have been unable to spread into this region."
The researchers attribute this slowing of evolution to the filling of ecological niches, or exploitable habitats or resources for new species to adapt to. The formation of new species is usually thought to involve three steps. First, a species expands across an environmental range. Then a barrier, such as climate change or a geographic event, causes the species to separate into distinct populations. Lastly, the development of reproductive isolation—the inability to interbreed—finalizes the speciation process. This cycle then repeats, creating the breadth of diversity seen in nature.
Price and his colleagues argue that the expansion of a range cannot occur if there are no ecological niches for a species to expand into. Despite the ability of birds to fly and cross geographic barriers, they cannot persist in regions where they are outcompeted by existing species who occupy available niches. In the eastern Himalayas, the researchers found evidence of this in numerous differences in feeding method and body size that appeared early in the evolutionary history of songbirds. Less dramatic ecological differences, such as living at differing elevations, appeared to form later as the initial adaptive radiation slowed.
"Our argument is that niche filling has stopped species from getting big ranges," Price said. "In the eastern Himalayas, it has become harder and harder for new species to get into that system, and we are quite close to the maximum number of species that can be accommodated. There is little room for more species because niches are increasingly occupied."
This model for diversification stands in stark contrast to previous hypotheses, many of which have focused on the slow development of reproductive isolation as the limiting factor.