First, let's be clear on what it is that's accelerated. Evolution is a change in genotype frequencies across generations, but that by itself can just happen randomly. Adaptive evolution, what this paper is about, is what you get when genotypes correlate with number of offspring—in other words it's the subset of changes in genotype frequencies due to systematic (i.e. nonrandom) biases in the genetic sampling over many generations.
Why should we expect this to have accelerated in the relatively recent past? What's changed in that time? The short answer is "radical population growth combined with new environments". Both of these changes are obvious in the sense that "everyone knows" they happened, but few people recognize their significance because they have no mathematical picture of how all the major variables fit together and interact.
Start by looking at the mutation rate, which we'll dub μ; this will be a tiny number in the neighborhood of about 10^-5 mutations per genetic locus per generation. Combine this with the number of individuals in the population, which we'll call N, and you get the number of mutations per locus per generation, Nμ. Obviously then, the number of mutations per generation scales linearly with population size. But the vast majority of these mutations with be non-adaptive—either deleterious or neutral. In fact, in a hypothetical population that's at a steady state of 1) no change in population size and 2) being at an adaptive optimum, there'll be no adaptive mutations at all.
Humans radically break both conditions of that hypothetical steady state: Our population has expanded by about three orders of magnitude since the end of the last Ice Age alone, and in that same time our environment has changed in many ways—new pathogens, new food sources, new climate conditions, new social dynamics, etc. When a population is knocked far from an adaptive optimum like we've been, the distribution of fitness effects for new mutations becomes skewed in such a way that the probability of a new mutation being beneficial goes up. In a hypothetical population at the lowest possible point on the fitness landscape, every new mutation is beneficial—when you're at the bottom, there's nowhere to go but up.
So we can introduce a hypothetical third variable which we'll call d, as a parameter measuring difference between the average number of offspring in a population and some hypothetical maximum value of that average. Ndμ would then give us the expected number of adaptive mutations per generation. (Bearing in mind that as a population undergoes adaptive evolution, d will shrink.) Another implication of this logic is that the further from an adaptive peak a population is, the bigger the effect size of the average beneficial mutation is likely to be. When you're near the top of a mountain any large jump in any direction is likely to land you further down, but when you're nearer the bottom a big leap can much more easily land you further up.
But the generation of adaptive mutations is only one third of the story. In order for evolution to be appreciable, a mutation has to spread through the population, and that takes time. If the number of generations required for a beneficial mutation to sweep through the population were also a linear function of population size, cranking up N would just be a wash—more adaptive mutations, but they take correspondingly longer to spread, so the overall rate of evolution stays the same.
But of course it isn't a linear function at all, which is obvious when you think about how reproduction tends to happen: One set of parents will often have several children, and if parents with a particular genotype tend to have more offspring then the spread of the allele responsible will be exponential. Time for a sweep to occur (measured in generations) turns out to be (2/s)ln(2N), where s is relative fitness advantage conferred by the mutant. So the speed of an adaptive sweep is linearly proportional to selective advantage and only logarithmically (i.e. strongly sublinearly) proportional to population size. When you crank up the N by three orders of magnitude, time to fixation doesn't so much as double.
So we have many more beneficial mutations being generated and not much longer time for them to spread—assuming, of course that they will in fact spread. Unfortunately, when a mutant first appears, the vast majority of the time it disappears soon after even if it's got a fitness edge. In a static population, the probability of a selectively beneficial allele sweeping to fixation is 2s, i.e. double its relative fitness. This means even a mutation with a 10% fitness advantage (which is abnormally large) has an 80% chance of being lost due to sheer randomness when it's still at a low frequency in a population.
The final piece of the acceleration triptych comes in recognizing the effect of population growth on the probability of a beneficial allele fixing: When a population is growing, the probability of a new adaptive allele sweeping to fixation becomes 2(s + r), where r is the population's rate of growth per generation. So if you've got growth occurring at a rate of 1% per generation, a new mutant with a 1% fitness edge then has a 4% chance of fixing rather than the 2% it would have in a static population. On the other hand, a mutant with a 10% advantage would only gain a boost from a 20% chance to a 22% chance. So population growth tends to help give the biggest boost to new mutations that have small but positive fitness advantages, and given the nature of the distribution of fitness effects (which looks like a power law distribution), most of the beneficial mutations will be in this range.
So we have environmental changes and population increases working together to cause a remarkable spurt in adaptive evolution. So says the theory, which is derivable straight from Darwinian first principles. The theory can only be wrong if our understandings of either human history or the principles of evolution are wrong, and neither is plausible. But we still have to go check it, and how Hawks et al did this will be discussed in a future post.