It was the language of the educated because of the Roman Empire. The elites used to study how to be Roman elites before the 5th century, and when Rome fell they just continued, as it was the established sign of higher class. The elite has been educated by Cicero for the last 2 millennia, and continues to be so.

A good analogy is French being the language of the elite in Britain, well after the Normans got merged into the general population.

They were written in Latin until well into the modern era. Newton wrote his major work in Latin, for example.

Because other languages didn't have a writing system, which is why every western language is using latin alphabet now.