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.