After the Western half of the empire fell in 476 that started the slow transition of the remaining eastern half to become mostly linguistically hellenized. The Eastern half of the empire’s lingua franca was always Koine Greek going back to the days when that part of the empire was incorporated into the Roman polity.

Latin was still spoken and used as a language of administration, law, and the military into the early 7th century or so. Especially in the areas of the Western Empire that were somewhat successfully reconquered for a time during the reign of Justinian the first. Justinian himself was a native Latin speaker.

However, the trend of the gradual Hellenization of the Roman Empire probably was really put into overdrive during the reign of Emperor Heraclius, after the epic titanic struggle of the last Persian/Sassanid-Roman War of 602-628. In which both empires had fought themselves to an exhaustive standstill, that depleted the resources and manpower of the Roman Empire and the Sassanid Empires (The Sassanids temporarily conquered large swaths of the eastern Roman Empire, almost recreating the territorial extent of the original achaemenid empire of Cyrus the great and so forth), after Heraclius managed to successfully General his way from kinda saving the Roman Empire from totally collapsing and fought the Sassanid Persians to a standstill, both empires were drained, and the ending settlement peace treaty between the two great empires basically returned to the status quo ante bellum territorially.

Only a few years later however, the Arab/Islamic conquests started out of the Arabian peninsula (The prophet Muhammad and Heraclius were contemporaries). The Rashidun Caliphate managed basically the perfect timing to start the rapid military expansion because the two local superpowers of the area Rome and Sassanid Persia were still crippled and barely recovering from the aforementioned almost 30 year war between them. The Sassanids also just came out of a civil war that happened just after the end of the last Roman-Persian war.

The Battle of Yarmuk was one of the landmark battles that helped shatter the still recovering Roman forces in the Near Eastern geographic region, and eventually the Caliphate swiftly conquered large swaths of the eastern Roman Empire and the entirety of the Sassanid empire due to these factors, and the respective very competent and effective military of their Arab opponents.

The Eastern Empire didn’t fall unlike their Sassanid counterpart, partially because Heraclius assessed that after defeats such as the battle of Yarmuk, that it was simply impossible to hold the Levant and Egyptian provinces, and later the other North African provinces that belisarius reconquered for Justinian. Heraclius had to abandon the region and pull all the way back to Anatolia (modern day Turkey) to use the Taurus mountains as a geographical bulwark/shield to save what was left of the Empire. Long story short is that the Arab Conquests left the Eastern Roman Empire a much reduced territorial rump, who’s remaining heartlands being the lower balkans, Anatolia, the Aegean Islands, and sometimes Sicily, Cyprus, and the southern boot of Italy left populated with only really, or at least predominantly Greek speakers.

Although I will also say that Constantinople was seriously besieged at least twice by the Arabs although in the late 600’s and the early 700’s but failed to take it much like the earlier joint Sassanid-Avar/Slav siege of Constantinople partially due to the Theodosian Walls serving as near impregnable bulwark as a masterpiece of defensive fortifications of the ancient world that allowed Constantinople to serve as an anchored redoubt against most opponents that attempted to siege the city for nearly a thousand years, until the advent of gunpowder technology and the invention of adequate cannons that allowed for the cracking of the shell of the Theodosian Walls in 1453, when the Roman Empire finally fell, and more generally rendered defensive wall city fortifications obsolete.

After the failed Arab siege of 717-718, the concerted attempts by the Umayyad Caliphate to take Constantinople, and therefore totally extinguish the Roman Empire ceased. However, there was still near yearly raids into Anatolia by Arab armies for decades and decades for plunder and capturing slaves and whatnot, with some lulls here and there. But still left much of the interior of Anatolia for a couple of centuries devastated and sparsely populated (the western part of Anatolia was the most densely populated and wealthiest part of Anatolia due to that area being on the Mediterranean Coast and being an area of fertile arable land and river valleys were for the most part not as touched by raiding and back and forth fighting as the interior of Asia Minor). The raids and counter raids between the Romans and the Umayyad Caliphate and its successor the Abbasid caliphate in Anatolia continued for a couple of centuries here and there, but the front was largely stabilized around the geographical mountainous redoubt of the Taurus Mountains. There was also significant naval and island based conflict in the Mediterranean as well with Muslim forces wresting Cyprus, Sicily and Malta from the Romans as well for very long periods of time.

So a combination of the invasion of the Lombards into significant chunks of Italy, and then later on the Franks/Carolingian (think like Charlemagne) empires to the west. The Avars, Slavs, and then the Bulgars in the northern to central Balkans in the north. And the various Caliphates and their respective successors after the fragmentation of the Abbasid Caliphate/empire in the east. Essentially boxed in and sandwiched the Roman Empire to a predominantly Greek heartland for many centuries, which gradually caused the linguistic and somewhat cultural hellenization of what was left of the Eastern Roman Empire.