Well, the term Dark Ages is out of fashion for the reasons in the post AgentEv2 linked. But I think your general question is how did Western Europe deviate so far from the Roman culture and control when the Empire was still very much alive in the East?
As early as Marcus Aurelius emperors realized that the empire had grown too large for just one person to rule. His attempt at a co-ruler Lucius Verus wasn't particularly helpful. And it wasn't completely necessary at his time but he definitely saw the trend that was coming. By the time of Diocletion it was clearly necessary, which is why he tried to split the empire with two Augusteses and two Caesars. The succession was often murky after that but the split between the West and the East was common after that. And the ruler in Constantinople was almost always supreme. It became clear that the cultural and power hub of the empire was now in the east. That's where people immersed in Roman culture gravitated.
At the end of the second century and then very much in the third The empire was facing pressures on its borders unlike any they'd seen in centuries. Now, most of these in the West weren't people who wanted to conquer Rome (in the east the Parthians succeed by the Persians had grander plans). In the west, they were often German tribes being pushed out by scarier people and wanted inside the safety of the empire.
More and more tribes began to be relocated within the empire in exchange for military service. Some of these tribes were treated horribly, like the Goths who were fleeing from the Huns, who then rebelled (the Gothic wars). More often these tribes would be formed into military units under the command of their chief or king. For example, Alaric the first king of the Visigoths who sacked Rome was merely looking to be made military commander in the West. He wasn't trying to conquer anyone.
Increasingly in the West, people who descended from Germanic tribes had key positions of power. You had a situation where a man like Stilicho, half-Vandal, was made a regent. After Stilicho, continuing until the very end Germanic leaders were the true powers behind the throne in the West. And they kept losing territory. First, Britain (who wrote begging for help from their constant invasions from the Scots and the Picts), then most of Gaul, and finally Hispania and Italy. Most of the Western provinces outside Italy, such as Gaul or Britain, never had libraries built on a large scale or any place where you could just pick up and read what the Romans left behind. And many of the places that did exist were destroyed.
Basically, the West was becoming less Roman. And the Eastern half of the empire was increasingly unwilling to invest a lot of time and effort into keeping it under the Roman thumb. This wasn't necessarily because they didn't care about keeping the empire as one. But the East had its own problems and succession issues. Eventually the entire fiction of there being a western empire was just quietly done away with. First by Odoacer and then by Theoderic the Great. (And, as an aside, the Gothic rulers of Germany were quite good and kept things running very well. It was Justinan's attempts to reconquer that destroyed Italy.)
So, the former provinces were ruled by new Germanic tribes (the Franks in France and some of Spain, the Angels, Saxons and Jutes in Britain, the Visigoths and Vandals in Spain, the goths and later the Lombards in Italy). These people had little to no relationship with the Eastern Empire. Mostly they were only just recently converted to Christianity or still pagans. A lot of them had no written script nevermind the ability to read Latin. They were just not primed to be culturally part of Rome, at that point. It wasn't a drastic fall. It was the cultural hub of Rome moving east over the course of a couple of centuries. And the west becoming increasingly Germanic. Until finally they just stopped ignoring reality. It was a slow process.
Once the break was complete (except for Justinian's attempts at conquest), the main cultural force was the Church (although it was not the political power it would later be). And the Church was not particularly interested in preserving pagan works or plays or poetry at that point. And if they didn't do it then no one would. The Church was a huge force in Constantinople too but it could never be the only source of learning and literature. This accelerated the split in culture. That being said the Germanic people did have their own rich culture and literary tradition. Sadly, so much of it is lost to us (we only just barely got Beowulf). But we do have the old Norse poems and stories, which are really quite wonderful.
Part of the reason the "Dark Ages" is a misnomer is because the wish to reclaim the cultural heritage of Rome came very early on. Charlemagne was very active in increasing education and transcribing ancient texts, some of which he'd been exposed to through Moorish Spain. Alfred the Great was similarly interested in this in England. Early on Christian scholars were trekking to Moorish Spain to copy manuscripts. Rome was a powerful force in the imagination of Western Europe long after the fall. It's why Charlemagne was made Holy Roman Emperor (also to piss of the Byzantines).