Translators and scribes busting hump for Charlemagne banged out copies of Latin classics during the (not so) Dark Ages, too. Over the next few centuries, Italian and French scholars translated Arab and ancient Greek philosophy, math, and medical texts as well. So by the 12th century, students in Western Europe had access to an extensive library of Classical and Eastern knowledge. The Dark Ages may not have had Google Books, but hey, it was way more than the "nothing" we've all been told.
But if all that ancient knowledge was there all along, why wasn't there a Renaissance sooner?
Actually, there was. Twice. The Carolingian Renaissance and the Twelfth-Century Renaissance happened before the Renaissance, though they're both largely relegated to footnotes in our history books. Why? Because they lacked the one thing that makes the Renaissance stand out: stupid pants. Wait, no -- they had plenty of those. What these proto-Renaissances really lacked was any significant, memorable advancement of the arts. And the reason for that was simple: money.
Thankfully, the groundwork was eventually laid for us to have both money and stupid clothes.
As the Middle Ages wound down, Florence's pockets were bulging thanks to its textile industry. By the 15th century, Florence was like Wall Street before the subprime bubble burst: lots of people getting rich, not by perpetually toiling but simply by moving income around. Merchants and bankers suddenly possessed the ability to make it rain like only kings, popes, and unhappy druidic weather-gods could before. They also possessed something entirely novel at this point in human history: the leisure time to appreciate things.