Through the use of interpreters.

In the case of the Spanish, they had interpreters with them all the time and in their interactions with the Aztecs and forming alliance with the Mexican states, their knowledge of the local language was critical for these diplomatic overtures. Without the alliances with the native Tlaxcala, Totonacapan etc., all enemies of the Aztec, the Spanish wouldnt have been able to advance into central Mexico and defeat the Aztec by themselves.

So translation worked through a chain. Cortés would speak Spanish with Gerónimo de Aguilar, who then translated into Yucatec-Maya for Malintzin, who in turn translated into Nahuatl for the Aztecs, before reversing the process. Malintzin was the "reference point" in the conversation and Cortés was called 'Malintzin' by the Aztecs.

As for the interpreters, Malintzin was a Nahuatl-speaking Nahua woman who was given as a gift to the Spanish and she was raised with them, and Aguilar was a Spaniard who was taken as prisoner by the Maya and later became a free citizen who learned the language. Malintzin spoke poor Spanish so Aguilar was used to translate it to her. These circumstances allowed for Cortés to comfortably navigate through Mexico.

The great thing about Nahuatl is that it was so widespread in Mexico at the time because it was spread by the Aztecs and became a lingua franca in the region and the language of diplomacy and trade, so knowledge of Nahuatl was very useful and most natives understood it. So when the Spanish sought to convert the natives, priests learned Nahuatl and they used Nahuatl-language texts to get their message across.

The same thing took place in South America, where the Spanish under Pizarro obtained a Tallán Quechua speaker named Felipillo who they brought back to Panama to teach him Spanish. Like the status of Nahuatl in Mexico, Quechua was a lingua franca of the Andes and the language of the Inca.

As for the first Europeans like Columbus, they decimated the Caribbean with war and less focus was given on understanding the natives and trying to coexist with them. In North America, the Northern Europeans like the English isolated themselves from the natives while the French took the Spanish course of action and tried to conform to the local culture and priests and fur traders learned the local languages by assimilating into the tribes. The only time Europeans knew absolutely nothing was probably with the Arawak in the first few voyages and Columbus famously brung along a Arabic-Hebrew interpreter thinking he was reaching Asia.