In general, the native Indian languages in Latin America were able to survive. This was because the colonial governments were not forbidding the native languages and were not punishing the people for using them. Instead, the politics was, that the language of the colonists is a must-known language for everyday communication between the different people, but they can use their own languages in the communication between themselves and at home. This is why the countries in Latin America nowadays have multiple official languages, most of which are native Indian languages.
To spread Christianity. The Jesuits were a big part of this spread too.
