Just like today, they had translators and interpreters! There was quite a lot of contact between the Mongols and Europe, both positive and negative. The crusader states in the Near East still existed, so there were already Latin Christian outposts in Asia. They already had lots of experience communicating with and learning the languages of people further to the east. They often weren't too interested in the Muslims who lived further east, but there were also ancient Christian communities out there, as far away as India and China, so there had already been missions there before the Mongols entered the picture. European Christians generally referred to the far eastern Christians as "Nestorians", which is pretty vague and inaccurate, but there were many Nestorians among the Mongols as well.

In 1245, Pope Innocent IV, sent some ambassadors to the Mongols. They met with the Mongol Khan, Guyuk, at Karakorum in 1246, and gave him letters from the Pope explaining the basic tenets of (Latin) Christianity and encouraging him to convert. Guyuk was not a big fan and responded:

"Thou thyself, at the head of all the Princes, come at once to serve and wait upon us! At that time I shall recognize your submission." (Allen and Amt, pg. 369)

A few years later King Louis IX of France also sent ambassadors while he was on crusader in the east, but the Mongols were not interested in him either, unless Louis was willing to submit to the Mongols first.

But how did everyone know what the other side said? They used intermediaries who understood both languages. The most common interpreters were the Armenians, who had been in contact with the crusaders since the time of the First Crusade, so they were already very familiar with Latin and French. They were also on the edge of the Persian/Arabic/Turkic world, so they often knew those languages as well, and in addition to that they were among the first groups of westerners to encounter the Mongols. The Mongols themselves sometimes used Persian as a lingua franca as well, although not always - the Wikipedia article you were probably reading (if I'm thinking of the same one) has a picture of a letter from the khan to the pope that's written in Mongolian.

Later in the 13th century, there were western Europeans living out in Asia as well who could act as interpreters. In the 1280s there were lots of Italians in the Ilkhanate in Persia, and there were Genoese colonies in the Black Sea that had contact with the Golden Horde.

Sometimes though, there were difficulties translating or finding competent interpreters. William of Rubruck, who led a mission to the Mongols in the 1250s,

"complained repeatedly of the inadequacy of his interpreter – a man, he assures us, ‘who was neither intelligent nor articulate’ and whose preferred method of translation, the friar discovered, once he had himself acquired a little knowledge of the language, was ‘to say something totally different, depending on what came into his head’, with the result that Rubruck chose rather to remain silent." (Jackson, pg. 264)

Rubruck also mentions in a letter that he and an interpreter were trying to explain Latin Catholicism to Nestorian Christian Mongols, and they just laughed at him. Even if he could make himself understood, it didn't necessarily mean they would care! Why should they submit to Rome when the Mongols were already the masters of the world? Obviously their own version of Christianity was superior.

Another missionary, John of Montecorvino, claimed that he had learned to read and write Mongolian himself. Supposedly he was able to translate some of the Bible from Latin into Mongolian. Missionaries to the Golden Horde also tried learning the languages themselves. Eventually back in Europe they realized that maybe they should start studying these languages *before* they set out on their missions, so language schools were established in the universities. But they needed people who knew the languages well enough to teach them, and learning Mongolian at a European university did not necessarily mean that the missionary could speak it well enough to explain complicated theology once they reached their destination. (Just like today, immersion in a culture is much different than learning a language in school!)

So, in short, western Europeans used interpreters from Asia and the Middle East who were already familiar with the languages, and then they tried to learn the languages themselves.

My main source for this is:

- Peter Jackson, The Mongols and the West (Routledge, 2005)

Other good places to look are:

- Igor de Rachewiltz, Papal Envoys to the Great Khans (London, 1971)

- Antti Ruotala, Europeans and Mongols in the Middle of the Thirteenth Century (Helsinki, 2001)

- David Morgan, The Mongols, 2nd ed. (Blackwell, 2007)

- Albrecht Classen, Multilingualism in the Middle Ages and Early Modern Age (De Gruyter, 2016)

There are some English translations of primary sources as well:

- Peter Jackson, The Mission of Friar William of Rubruck: His Journey to the Court of the Great Khan Möngke, 1253-1255 (Hakluyt Society, 1990)

- Peter Jackson, The Seventh Crusade, 1244-1254: Sources and Documents (Ashgate, 2007)

- Malcolm Barber and A.K. Bate, Letters from the East: Crusaders, Pilgrims and Settlers in the 12th-13th Centuries (Ashgate, 2010)

- Jessalynn Bird, Edward Peters, and James M. Powell, Crusade and Christendom: Annotated Documents in Translation from Innocent III to the Fall of Acre, 1187-1291 (University of Pennsylvania Press, 2014)

- S.J. Allen, Emilie Amt, eds., The Crusades: A Reader, 2nd ed. (University of Toronto Press, 2014)