How did people from different cultures communicate to each other? For example, how could the European explorers meeting the native peoples of North America come to learn each other's completely different languages?

This is a very specific example, but the Spanish had a slave known now as "La Malinche" who understood several languages and could usually translate directly for the Spanish or they could find somebody who could translate for Malinche and then she could translate for the Spanish. She spoke Mayan and Nahuatal. Bernal Diaz del Castillo who wrote one of the most detailed first person accounts of Cortes' exploits, said "Without the help of Doña Marina (La Malinche) we would not have understood the language of New Spain and Mexico." (Díaz).

Thomas, Hugh. Conquest: Montezuma, Cortés, and the Fall of Old Mexico. New York: Simon & Schuster, 1993, pp. 171–172 Diaz, B., 1963, The Conquest of New Spain, London: Penguin Books,