In this answer, I'll be considering the native groups from South America, more specifically in Brazil, during the XVI-XVII centuries.

The Portugueses arrived in Brazil in 1500, but only started to colonize the region around 1530-1600. They encountered a enormous ethnic and linguistic variety among the natives, with over 700 indigenous languages. One in special, used by the Tubinambá nation, was more common in the Brazilian coast, due to the variety of groups that had relations with the Tupinambás. Therefore, the Europeans, colones and missionaries realized that they had to learn that language - the "Tupi" - in order to achieve the colonization project, given that it was only possible through stabilizing communication with the natives.

They started to learn Tupi, that became the language spoken by the Europeans and Native groups and essential to the maintenance of colonial control on the territory and it's people. By the XVII, Tupi was considered the "general language" (or "Língua Geral"), and even had regional variety, the Amazonian General Language (spoken by the Amazonian native population and the Europeans, and was used until the first half of the XIX century) and the Paulista General Language (used by the Natives and Europeans more to the south of the colony).

But how exactly the Europeans learned how to speak Tupi? In the beginning, the leaning process was more through the conviviality and cultural immersion among the native. Later, the missionaries that worked for the Portuguese started to "reduce" the Tupi Language from a Oral-based language to a Written-based language, using the Latin Grammar as a model.

It is important to notice that, by the time, Religion and State Power were bound together, specially in Portugal and Spain. Therefore, in order to fully control the new territory, they used the missionary action to systematize and learn the native languages. Catechism and Territorial Control were directives that bounded the Church and State in the "New World".

In Brazil, the missionary order Society of Jesus had a fundamental role in that process. Being a catholic order, the primary goal was to convert the Native, and, in order to do that, it was mandatory to every newcomer in the Society to learn the General Language, so they could preach in the native's own language. In a instructional document written by a Jesuit priest c.1650 to all the missionaries working in the Amazon region, there is a note to the preaching process when the missionary, or the native, can't speak the General Language: "They [the missionaries] will show them images of saints and crosses, and will make them watch the divine oficies, the administration of sacraments and other christian actions, in order to, in case of necessity, teach the baptism to the by mimicry" (in a free translation).

Later on, by the end of the XVII century (almost a 100 years from the start of the learning process), there were grammar and dictionaries of the Tupi language, with translations to Portuguese, French, Old German and Spanish.

In short, during the Portuguese colonization in Brazil, the Metropolis used the missionaries order's works (mainly the Society of Jesus) during the second half of the XVI century and the entire XVIII century (1530-1700) in order to learn, systematize and use the native languages to make the whole colonization process easier. Starting with mimicry and pointing to images and ending with fully systematized dictionaries and grammars, that process took roughly 150 years. And, of course, as learning is a two way process, as the Europeans learned the Tupi, the Tupi speakers learned european's languages. There were, as well, schools in the villages run by the Jesuits, where they taught portuguese for the Natives and Tupi for the non-speakers, Europeans or natives.

By 1758, the teaching of the General Language or any other Native Language was prohibited by the Portuguese Crown Prime Minister, the Marquis of Pombal, and every citizen of the colony was supposed to learn portuguese, even the Natives. That's why, for short, why Brazilians in general don't speak not even one of the over 700 natives languages that existed in the beginning of the colonization process, apart from the few Native populations that survived.

I hope I have helped you in your question. I'm from Brazil, more specifically from the Amazon, and study the colonization of the Brazilian territory by the portuguese in a cultural perspective, which includes the language.