Liturgical language is always going to be affected by both scriptural language and spoken language. In the case of the Christian sacred texts, about two thirds are written in Hebrew, the most cited parts are written in Greek, and a handful of pages and isolated quotes are in Aramaic. Meanwhile, in the formative years of Christianity, the West of the Christian world spoke Latin, the East spoke Greek, and the East beyond the Empire spoke Aramaic.

And the fact is that Aramaic is the liturgical language of certain parts of the Christian church, which are located, as you might guess, in the east of the Empire. I'm thinking particularly of the Assyrian Church.

But the other parts of the Empire used their own vernacular as their liturgical language. The Greek parts used the Greek New Testament in its original form and used the old Greek translation of the Hebrew Bible known as the Septuagint (or LXX). The Orthodox Church that descends from the Eastern Empire's church does so (with some national variation) to this day.

The Latin part of the Empire translated the Bible into a staggeringly huge variety of Latin translations, which were eventually cleaned-up and codified by Jerome into the Vulgate translation. With some room around the edges, this translation is the basis of Roman Catholic interpretation to this day.

In other words, the question is incorrect in its premise: Latin isn't the liturgical language of the entire Christian faith. It's the language of the West, which eventually became the dominant branch of Christianity despite a weaker start. In fact, Greek probably has the best claim to being the common language of Christians; the word Christian is basically a Greek word itself. The distinctive texts of Christianity were furthermore written in Greek, and Greek words pop up even in Latin-influenced liturgy ("kyrie eleison.") Finally, as Biblical scholarship improves, Christian clerics, scholars, and educated laypeople around the world understand basic Greek words and concepts such as logos, agape, kairos, ktl.