Original question from Quora:
My thesis work was in Latin American studies, and as such I read a lot of the “first encounter” diaries and biographies that were written. The answer is actually surprisingly easy: CHILDREN.
Columbus, Cortez, and likely many of the other explorers kidnapped young children or took them on as servants. As children learn language incredibly quickly, it was not long before they could work as interpreters for the Spanish. By the time of the second voyage to the Americas, Columbus had fluent interpreters.
Other accounts of first meetings between the Spanish and the Native Americans show similar patterns that work from there. The Indian tongues could be divided into several groups, but even if you had an interpreter from one Indian language, that person could communicate with other tribes not too far off, much the way Spaniards and Italians could (and still can) communicate even though they don’t speak exactly the same language.
When no interpreter was available, they were reduced to drawing pictures and using gestures, just as you would imagine. However, language is learned rather quickly, especially when you have a child’s brain to help you!