From the book A Distant Mirror, Europe bounced back relatively quickly after the Black Death and the political collapse of the 14th century. What in the events of the 14th century helped permit rapid advance in Western European civilization in the 15th and 16th centuries?
Innovation and the loss of skilled agricultural workers were, in my opinion, the two largest factors here. With too many "nobles" and too few "serfs," the system began to break down, creating a gap for true innovators. Note also that education was beginning to increase; the printing press allowed more and more people to understand how culture and government worked, and feel embittered about being forced to work for elites while getting little in return.
#4 makes a number of great observations about the way in which the death of so many peasants resulted in the grounds for change and good social advance to occur. The way in which so many died meant that feudalism took a massive blow, as the surviving peasants were lucky enough to be in such high demand that they could protest about their conditions and bring about change.
I would say that one of the great catalysts for development was the East. Constantinople was still in tact and when intellectuals moved over to the West, they brought learning, arts, and so much more. When the Turks sacked Constantinople, then more intellectuals came to Italy and this made a lasting change and brought development.
A Distant Mirror, as many reviewers of the time, and indeed Tuchman herself noted, was rooted in her assumption that society in her own time was breaking down. Tuchman views the period as maked by senseless violence, duplicity, and ignorance. She does not engage with much of the scholarship of the period, and, as many historians of medieval history noted (see the American Historical Review vol 84, No. 3 if you can get access for a famously scathing assessment by medievalist Bernard Bachrach) doesn't seem to have much of an idea of what made medieval people act as they did.
As for the emergence from this period of barbarism, as she portrays it, I thoroughly agree with pohnpei's assessment: she doesn't seem to have much of an idea of why this happens, other than her often-repeated faith that history just moves, and sometimes "ideas" just break out of "molds." She's not interested in thinking about causality, just in using the fourteenth century as a "mirror" of her own time. There are lots of problems there. That said, I have used this book in European history courses as a way to disabuse students of any notions of "knights in shining armor" that they might have, but I wouldn't use it as an example of how to do history. Guns of August, on the other hand? Different story.
Usually catastrophes like the Black Death would wipe out a civilization; in this case just the opposite happened -- by the 1400's the Renaissance had begun. Part of why Europe bounced back was due to the economic changes that had been occurring, but which were suddenly accelerated due to mass death. Labor became expensive, which forced innovation and improvements in agricultural technology. Feudalism suffered a fatal blow, as the peasants were free to negotiate wages wherever they pleased; no one continued to be bound to the soil. Displaced farm laborers in time became craftsmen; guilds became prominent; trade expanded, wealth increased, and the goods and services provided stimulated interest in the arts and sciences, and brought the Renaissance.
Interestingly, I just finished reading that book and I didn't get much from it about rapid advance. It seemed like a book that was very negative about the 14th century with its constant discussion of how incredibly greedy and shortsighted the aristocracy was.
At the very end of the book, she talks about how Protestantism was coming and so was more centralized government. She talks about how technology was about to make the "discovery" of ways around Africa and of the New World possible. She talks about how the printing press was just being invented. I suppose these kinds of things could be what allows the rapid advance. But Tuchman seems a little less certain. She ends the book talking about how things got worse after the end of the centurey
...until at some imperceptible moment, by some mysterious chemistry, energies were refreshed, ideas broke out of the mold of the Middle Ages into new realms, and humanity found itself redirected.
Doesn't sound to me like Tuchman has a clear argument about the causes of the advances.
One explanation offered is that most of the people who died were elderly and infirm. This is not often the case, for instance, in examples of devastating warfare, such as World War I, or even in the case of a more recent outbreak of disease: AIDS. Interestingly enough, one of the earliest studies of this topic was done by the RAND corporation in the mid-1960s and came to the conclusion that disasters (such as nuclear war) might not be as bad as had been earlier imagined!
Here's a good overview of recent research on the black death: