In chronological order:
1. A garbage dump. The garbage dump outside of Oxyrhynchus, which was a city founded in Egypt after Alexander conquered the area in 332 BC and abandoned after the Arabs conquered it in AD 641. For the nearly 1000 years in between, people lived in Oxyrhynchus and threw garbage into big heaps outside of town. This garbage included papyrus with stuff written on it. Most ancient papyrus with stuff written on it has rotted away long ago, but some has survived because it was put into jars as in the cases of the Dead Sea Scrolls and the Nag Hammadi library, or into coffins with dead people, or, in the case of these garbage heaps at Oxyrhyncchus, because the climate just happened to be just exactly right. A huge amount of papyrus was recovered there in the late 19th and early 20th centuries. A little over 5000 pieces, a small fraction of the total, have been edited and published so far, including many copies of existing and previously-lost Classical Greek texts and a few very important for the study of Classical Latin.
2. Pope Gregory the Great. Important in a bad way: on his watch (he was Pope from 590 to 604) much of Classical literature went missing. In the case one Classical author after another, we have records of their being known, such as quotes or other mentions, up until the late 6th century. Did Gregory intentionally destroy all copies of Livy which came into his grasp? I can't prove that he did, but it doesn't matter. He was far and away the most powerful man of his time. He thought that the End was Near, that Hell was full with the souls of sinners and volcanoes were places were Hell was spilling over, and a lot of Classical literature, and competency in the Greek language, disappeared on his watch. Intent or incompetency, who cares? He's guilty, case closed.
3. Petrarch. Perhaps many of you know him as one of the three first great writers in Italian, along with Dante and Boccaccio, and that's fine and all, but nevermind that because Petrarch, in the 14th century, also started the Renaissance. Many people all along, all through the Dark and Middle Ages, had made heroic efforts to preserve the great literature of ancient Greece and Rome -- mention must be made of Cassadorius, who lived around the same time as Gregory and preserved much of the ancient literature Gregory destroyed either by intent or neglect -- but Petrarch is the greatest of them all. Many of the best manuscripts of ancient Latin literature we have today are copies made by Petrarch.
4. The 19th century. There actually seems to have been an increase, in the 19th century , of the number of people who studied the Classics. Many a 19th-century author writing in a vernacular quoted copiously from the Latin Classics, and didn't bother to translate, assuming that his audience was fluent. A few even assumed the same with Greek.
The recovery of texts in palimpsest, begun in the late 18th century, really got rolling in the 19th, with Cardinal Angelo Mai, librarian of the Vatican, leading the way.
I'm sure many of you have heard of the Oxford Classical Texts, begun late in the 19th century. I wonder how many of my non-German readers realize that the Teubner series, begun in the mid-19th century, is what the Oxford Classical Texts want to be when they grow up. The Oxford series is a wonderful thing, but it was begun in conscious imitation of Teubner, and Teubner continues to be the standard, with the largest numbers of titles in print, in volumes of the highest standards of construction.
They really are nice, you should check them out.
5. The Internet. Do you remember how, in the late 20th century, so many people predicted that technology would accelerate the dying-out of the more obscure languages? It has done the opposite. Remember how, in the early days of the Internet, it was predicted that languages not written in Latin letters, such as Greek, Russian, Arabic and Chinese, would be pushed out by technology? They learned how to format those languages, though, didn't they? No change of browser required any longer.
In the case of the Classics, there are wonderful online resources such as the Bryn Mawr Classical Review, the Rheinisches Museum and What's New in Papyrology, to name just a few.
6. The relentless onward march of technology. Like multi-spectral imaging, with which texts on papyri and parchment which had been considered unreadable because of wear and tear, dirt or overwriting suddenly come forth into clear view.