Latin superseded Greek in the West through a "clientele change", so to speak. The knowledge of Greek itself began to die out among the Westerners after the fall, contributing to some of the challenges that would lead to the Great Schism.

Greek was the common language of the eastern half of the Roman Empire whereas Latin was the common language in the western half. As Christianity began to emerge in the eastern provinces, it enjoyed a large Greek speaking community that progressively moved more and more to the West. Due to a strong vernacular difference in this half of the empire, Latin began to take prominence over Greek, at which point the Western Church made the shift from Greek to Latin in regards to the liturgy. When exactly did this shift occur? It's rather difficult to pinpoint an exact date, though there was a transition period from the 3rd Century where we see some of the last written manuscripts of parts of the liturgy in Greek to a completion phase around the 4th or 5th Century where the Mass was completely Latinized. Pope Gregory I would edit the Liturgy one more time around the onset of the 7th Century and the Roman Mass would remain virtually untouched for over the next thousand years. As Latin was the vernacular, the usage of Greek began to die out in the west, especially after the collapse of the Western Roman Empire. Many of the Western clergy were also simply not educated in Greek, a prominent example being that of Augustine of Hippo.

Another thing to bear in mind is that making Latin the official language was just as much a practical decision as it was done for traditional reasons. Latin, as a dead language, is stable and many of the words have meanings and understandings that predate our modern connotations. Latin provides an unbiased language through which all members of the Church could understand something and this allowed a more, open intellectual exchange, even if the writers had different native languages. Some important examples of this would be like Martin Luther and his exchanges with John Fisher and Erasmus or Petrarch writing to Cicero.

Most of my information comes from the various books, articles and pages I've read about the liturgy. I don't have many scholarly books to quote from, but I can list some of the sources that come from the Catholic Church and articles.