What in the 14th and 15th centuries permitted the rapid advance in western European civilization in the 15th and 16th Cent?
Europe bounced back relatively quickly after the Black Death and the political collapse of the 14th Century.
Actually, Europe did NOT bounce back quickly, the process took over 200 years. Also, contrary to the above answer the Renaissance was not "pretty remarkable," but was rather a natural response to events.
The first event which led to the rapid advance of European civilization was the Fourth Crusade. The Crusader Knights had been funded by Venetian Merchants who transported them to Constantinople for free. While there, the Knights became involved in a dynastic dispute which led to the sacking of that great city by European Knights--a disaster of hideous proportions. Many of the treasures of Constantinople were carried back to Italy, as a result of which Venetian merchants grew tremendously wealthy. This wealth allowed Venetian merchants to exercise a monopoly over Asian trade, in which wool from Europe was traded for silk, jewels, etc. from Asia. The intense wealth of these enterprises provided the funding for the artists and scholars of the Renaissance, none of whom worked for free.
A second, closely related event, was the fall of Constantinople to the Turks on May 29th, 1453. The fall of the city was largely the result of the damages caused by the Fourth Crusade from which it never recovered. Many scholars of Constantinople travelled to Europe where they served as clerics and educators. They brought with them the wisdom accumulated in Constantinople. Contrary to the above answer, they did NOT flee the city prior to its collapse. Unlike their Christian counterparts, the Muslims who captured Constantinople did not engage in wholesale slaughter and destruction; however many scholars believed their best interests lay elsewhere, and departed accordingly.
The combination of renewed scholarship and large amounts of money to fund artists, scholars, etc. gave rise to the Italian Renaissance. Francesco Petrarch was the first person to speak of a new age, and use the term "Renaissance." Still, it was not a "bounce back." The process was slow and arduous.
This is an excellent question. The 14th century was a complete disaster for the West. The problems that they face is almost unimaginable from our perspective. The Black Death alone was catastrophic. In light of this, it is pretty remarkable that that the Renaissance started.
While there is not one explanation for this, I think there are two important considerations.
First, we need to realize that Constantinople was still standing. And before the Turks sacked and took over the city, there were many intellectuals that fled to the continent. Many of them happened to settle in Italy, where the Renaissance started. Hence, the glories and the continued learning of the the Eastern Roman Empire made its war to Italy.
Second, there is something about death and disaster than can sharpen a person's will to survive. For example, constant warfare in the Hellenistic world was one of the catalysts for technology. In like manner, there was something in the human will to survive that we need to consider.