You're thinking of this book, How the Irish Saved Civilization? I read it a long time ago, and while I liked it then (I was going through my Irish American phase), I'm not thrilled with it now. It's very Western centered.

First of all, plenty of people following Peter Brown challenge the notion of a collapse of the Western Empire. The "barbarian" states often had strong connections to the Emperor in Constantinople (the Vandals in N. Africa are a good example, minting coins in the name of the emperor - of course this was probably a convenient fiction for both sides, but still, plenty of Roman culture continued on for quite a while)

Second, they did a terrible job of saving a lot of it. Greek became almost extinct in the west. I think the first school of Greek in Italy popped up only in the 1300s? And a lot of Greek texts were preserved in Constantinople. I don't think they had any Plato in the west, and only Aristotle through Arabic translations done in Spain. As for Latin, there were important libraries in the Vatican, Paris... I just grabbed my edition of Livy and I see manuscripts from Verona, Oxford, Rome, Leiden. Admittedly many of these are 9th and 10th century copies, but even if the archetype did only survive in Ireland, a lot of people in Europe after just a couple hundred years were interested in collecting ancient texts, had the ability on some level to read them, and the technical skill to copy them.