Justinian is considered to be a last Roman emperor to speak Latin as a first language. What does this fact imply about the changes in the Roman empire at that time?

This implies that the Western empire had ceased to be truly "Roman," as it had been taken over by Germanic tribes. And in the Eastern empire, Greek language proved more long-lasting than the Latin language.

I might add a little detail about the Greek and Latin languages in the Roman Empire. Ever since the conquests of Alexander the Great in the 4th century BC, the Mediterranean world in general experienced "Hellenization." Or we might call it "Greekification." The koine (that is "common") Greek language became a common language throughout the lands where Alexander had cast his influence. When the Romans took over those lands, Greek remained a commonly spoken language. So Latin was the official language of the Roman empire and Romans themselves were native Latin speakers. But Greek had become the language of international commerce. When the New Testament was written in the early AD era, it was written in Greek for that reason.

So, in the Western Roman Empire, Latin remained spoken and was the official language of government, but with Greek spoken alongside Latin in the marketplaces and cities. In the Eastern (or Byzantine) empire, Greek was predominantly spoken, while Latin remained known by educated persons, especially in government. And Greek did become the official language of the Byzantine Empire by the time of emperor Justinian. Justinian's famous law code, the Corpus Iurus Civilis ("Body of Civil Law") was published in both Greek and Latin editions.