In short: Administrative pragmatism meant that Latin never took as much hold in the east as it did in the west.

The central influence of Rome was constrained by a fundamental difficulty of premodern logistics and communication: When the fastest way to deliver a message on land is “messenger on horseback” and the only way to move large armies is along major routes in well-populated territory you can plunder “acquire” food from, the ability to effectively administer a region diminishes with distance. Naval transport and supply can extended that range, to the degree that territories are navigable. Thus, it is essential to have local structures to facilitate administration.

In western and middle Europe, such structures were largely absent, so Rome had to build its own, more or less by transplantation of Roman (educated) citizens via land grants to soldiers, creating a local military reserve and elite that could enforce Rome’s laws. However, Rome rarely bothered to stamp out local customs or enforce a strict cultural dominance: Whatever kept the locals content and compliant was good enough. This wasn’t progressive respect or inclusion or anything, just a pragmatic way to save on manpower.

That led to a form of bidirectional cultural blending: Romans would adopt customs from those around them, while those seeking social status would adopt customs and learn the language of the Roman administration (particularly if they hoped to become citizens themselves). This blending of cultures and identities meant that, even as Rome’s central power crumbled, the emerging local elites still spoke Latin, or some locally flavoured variant thereof.

In the east and south, the Macedonian conquest and successor states had already established an administrative system, staffed with a strongly separated Greek-speaking upper class. The same pragmatism that had led to the laissez a faire approach in Europe led to instead taking over the existing system, including Greek as dominant administrative language.

The Roman Empire was eventually separated into two sub-empires, each with their own subordinate emperor, because a single administration became unfeasible due to various conflicts and the sheer size of it. As the western half started collapsing, the eastern empire almost exclusively used Greek for official matters, eventually supplanting local languages. However, it had to contend with (and lost some land to) the Arab Empire, which brought its own family of semitic languages, then eventually fell to the Ottoman Empire, which also brought its own Turkish languages.

To sum it up, the reason it spread across Europe is the fact that it was a valuable language to learn for social and political reasons, while the reason it didn’t spread in the east is the existing of structures that made learning it obsolete, and its stand-in (Greek) didn’t survive the assault of other, powerful empires with enough administrative capacity to impose their own language.