In short, because medieval Christians were not the paranoid philistines of modern myth.

First, we should note that the "Eastern Church", meaning the Orthodox, didn't exist as an organism independent from the western Church until the Great Schism of 1054. By that point, the pagan classics were long since entrenched in both eastern and western European thought, literature, and culture. The study of how later generations viewed, reacted to, and adapted the classics is called classical reception studies, and there are reams of works written on the Middle Ages in particular.

Looking at the first few centuries of the Christian period will help us understand why these works survive. Despite the attacks of many pagan authors, like Celsus, on Christianity as a religion of the ignorant - and the fact that it certainly did draw a lot of poorer and uneducated to its ranks - Christianity drew a number of proponents from the educated upper echelons of Mediterranean society. Ambrose of Milan, for example, was the son of a Roman official and educated in Rome itself. He had a major influence on Augustine of Hippo, who was similarly educated in Classical literature. In De Doctrina Christiana (On Christian Teaching), Augustine introduced the extremely influential idea of "Plundering the Egyptians". Drawing on the story in Exodus 3 of how the Hebrews gathered gold and silver from their pagan neighbors before fleeing Egypt, Augustine formulated the idea that pagan works could have real value for Christians. In this he was influenced by Romans 15:4, in which Paul famously states that "everything that was written in the past was written for our learning".

A concrete example might help illustrate how this was done. Virgil attained great renown in the Middle Ages as the exemplary learned man - there's a reason he leads Dante through hell and purgatory in the Commedia - in part because of the erudition on display in the Aeneid, but also because of his Fourth Eclogue. It's a weird poem and even now no one really agrees what it's about, but it foretells the birth of a divine boy child who will act as a kind of savior and establish dominion over the world. It doesn't take much to see why later Christians saw this as a prophecy of the birth of Christ.

Another good example, and more standard, is the Ovide Moralise. This is a very long medieval work which acts as a commentary on the myths as presented by Ovid. Ancient and medieval Christians (and pagans) had a great fondness for the hermeneutic process of allegory; that is, reading a story non-literally to derive a moral or message from it. Paul applies this to the Bible in Galatians 4 when he reads the story in Genesis of the sons of Abraham allegorically, and it was a standard way of reading the Bible for millennia (and still is in many denominations, like Catholicism and Orthodoxy; strict textual literalism is a mostly post-Reformation phenomenon). The Ovide applies this to Ovid, and reads the myths allegorically - as containing important messages when read in a Christian framework.

That's not to say the myths weren't read for their own sake and pleasure; they certainly were. There was also pushback against this at times - the Anglo-Saxon bishop Alcuin famously demanded "What has Ingeld to do with Christ?" (Ingeld being a legendary hero of the pagan Anglo-Saxons and Norse). But by and large, medieval Christians held the old stories in immense esteem; Gregory Heyworth, in his entry on Classics and Mythography in the Medieval Encyclopedia, writes that "Medieval culture in Europe rests upon the twin pillars of the Bible and classical mythology."