As culture Spread across trade, so did medical knowledge.  The Greeks made earlier advances in medicine.  Working with study of Greek texts, Islamic society developed more medical knowledge through people like Avicenna.  There were many factors spreading this knowledge throughout Eurasia.

Crusades & Al-Andalus

During the Crusades, Europeans came in contact with Islamic culture including medicine.  Similarly During Al Andalus, Spaniards came in contact with Islamic medicine.

Improving European Education

Simply coming in contact with medical books in Arabic was not enough to spread the knowledge.  It took serious study and translation skills.  Europe in the early Middle Ages lacked much literacy beyond monasteries, but as time went on Europeans developeded a network of cathedral schools and the first universities.

Mongol Medicine

The Mongols were part of a wider network of Eurasian people who had developed a medical system of their own, including the Chinese, Korean, Tibetan, Indian, Uighur, Islamic, and Nestorian Christians. They took the medical knowledge of these people, adapted it to develop their own medical system and at the same time organized an exchange of knowledge between the different people in their empire. On their journeys throughout Asia, the Mongols brought with them a team of doctors. Usually foreign, these doctors themselves had brought medical knowledge from other people in Asia to the Mongol court. They serve three purposes on the journeys on which the accompanied Mongol princes. Their first purpose was to be the personal physicians of the princes in case they required medical attention. The second was to observe and obtain any new medical knowledge from the various groups of people that they encounter. Finally, they were to also spread the medical knowledge that the Mongols had put together to the peoples they encountered. The Mongols were also able to contribute new or more advanced knowledge on topics such as bone setting and treatments of war wounds because of their nomadic lifestyle. The Mongols were the first people to establish a link between diet and health.  (Source this section only: https://en.wikipedia.org/wiki/Traditional_Mongolian_medicine)