A college professor of mine is teaching us that in the middle ages in Europe people didn't move around, that there was no thirst for discovery, and that map making became rare and geographical knowledge was lost. Is this true? How could an entire continent stay so immobile?

In terms of journeys and exploration specifically, there was certainly interest in travel. The Middle Ages was when some of the first literature that we would recognise as 'travel writing' became popular. There were of course people like Marco Polo or Ibn Battuta who did these grand tours of the eastern world, recording many amazing and incredible things. But they were hardly alone in that. A few examples that might be worth chasing up if you're interested in some medieval travel writing:

Ibn Jubayr - a pilgrim from 12th century Iberia who documented his Hajj and visits to other holy sites in the Middle East. He's a particularly useful source for daily life in the Crusader States, as he often commented on the state of the lands he travelled through or odd events he encountered. For instance, he tells us about a wedding he saw, and a skirmish between Christian and Muslim patrols that was interrupted by a trade caravan passing through.

Benjamin of Tudela - A Jewish merchant and traveller also in the 12th century. His Travels of Benjamin were rather popular, and narrated his journey through various Christian and Muslim lands. He was particularly interested in the state of Jewish communities in the cities he visited, as well as how different peoples treated the Jews. He is one of our main sources about the ethnography of the 12th century Mediterranean.

Petachiah of Regensburg - A Jewish merchant and Rabbi at the end of the 12th century looking for overland trade routes to the east, most likely to link up Jewish communities living there. His journey took him from his home in Germany deep into the mountains of Iran, and then back again. Like Benjamin of Tuleda, he was particularly interested in the treatment of Jews by the various groups he encountered.

William of Rubrick - A missionary dispatched to the Mongols by King Louis IX of France. His report on Mongol culture and geography is a major source for what Mongol culture was like in the 13th century. Particularly notable is his contact with Buddhists, who he was asked to debate as part of a competition held at one of the Mongol courts he visited.

Giovanni da Pian del Carpine - Diplomat, archbishop, and missionary sent to see the Mongols by Pope Innocent IV. He got there a little before William of Rubrick and was probably one of the first Europeans met by a Mongolian Khan. He was going there to politely ask the Mongols to please stop invading Christian Europe, to which the Mongols replied that they were the scourge of God and all Christians should submit themselves to the Khan. As a diplomatic mission it achieved exactly nothing, but as a historical source it was very useful!

Odoric of Pordenone - Early 14th century missionary that went to see the Mongols etc. with a few other missionaries. He then continued into China and India before returning home.

I'm sure you get the picture by now. There was definitely movement and thirst for discovery, and explorers who wrote down their experiences. The motivation to explore came from a few main sources: missionary activity, pilgrimages, merchants trying to find new trade routes, and medieval people going "What's a Mongol?". These are just some of the people who wrote memoirs about their international travels. There were also more local travel writers, such as Gerard of Wales, an English bishop and royal clerk who wrote numerous pieces of travel writing about the British Isles. His Description of Wales, Journey through Wales, Topography of Ireland, and Map of Wales were all pretty popular.

However, there was very little exploration of Africa. But this wasn't for lack of trying. For example, in 1291 Vandino Vivaldi and Ugolino Vivaldi launched a two ship expedition to explore the waters off the coast of west Africa. They were hoping to find new trading partners for their city of Genoa, which was losing business to Venetian merchants. To say that "geographical knowledge was lost" is technically correct - medieval Europeans forgot the exact location of the Canary Islands until around 1400 - but that's about all the geographical knowledge that was lost. They followed the African coast down past the Canary Islands (though it's not known if the Vivaldi brothers saw them), then disappeared. This is because the waters in that part of the world are vicious, and there was no ship design that could reliably traverse those waters. The Vivaldi brothers were using galleys, so they were toast. Even as the Portuguese began exploring the coast of Africa in the later Middle Ages with sturdier ship designs, Portuguese explorers summed up how they felt about sailing that region when they named it 'Cape No'. Europeans certainly wanted to explore sub-Saharan Africa, and medieval literature contains quite a bit of speculation on what fantastical peoples must live there, but the combination of vicious seas and a massive desert were a bit much for European explorers until toward the end of the Middle Ages.

Some further reading:

Legassie, Shayne. The Medieval Invention of Travel. University of Chicago Press, 2017.

Cohen, Jeffrey Jerome. "Pilgrimages, travel writing, and the medieval exotic." in The Oxford Handbook of Medieval Literature in English. Oxford University Press, 2012.