While it may appear that the Catholic Church no longer uses much of the Latin language, the truth is much more complicated.
In fact, it was the Second Vatican Council (Vatican II) that promoted the use of Latin in the liturgy.
While the use of the vernacular language in liturgy was highly promoted, it wasn't meant to replace the use of all Latin in the Roman Rite.
In the Council document, Sacrosanctum Concilium, it states quite clearly what is to be done with the archaic language.
The vernacular was designed to be used for "readings and directives, and to some of the prayers and chants" (Sacrosanctum Concilium, 36).
The original intention was to replace many parts of the Mass with local languages, while preserving the rest in Latin.
This same document even recommends teaching the people how to sing in Latin!
The General Instruction of the Roman Missal similarly echoes these words, but puts them in the context of maintaining unity among different language groups.
The Church used Latin for centuries because it was born in the Roman Empire and when that empire fell, it kept it as a way to unify Catholics around the entire world.
Even many Church documents are still translated into Latin, and recently the Vatican launched a Latin radio station! This is done to preserve the ancient language and to help keep it as a unifying force for all Catholics.