This phenomenon is well explained in ``God's Playground: A History of Poland'' by Norman Davies, which I am currently reading/listening to. But historically Latin was the lingua franca that time, and Polish people who wrote were always educated. So no other possibility back day I'd say...
Yes, Latin was indeed the lingua franca of Europe then, but the situation is even more interesting here.
1. Poland at the time was an expansive, multi-ethnic state, and while Polish gained increasing dominance as the lingua franca of the state (other languages of the state administration included German and Ruthenian), Latin was for a long time the lingua franca even just within the Polish state itself.
2. Unlike other countries where education was concentrated exclusively in cities, Poland also had a dense network of parish schools that diffused knowledge of Latin among even the rural nobility and town-dwelling population. Later, there was also a network of Jesuit colleges that followed the Ratio Studiorum which included extensive education in Latin and made an elite education accessible not just to wealthy magnates, but to poorer nobles as well. Recall that the Polish szlachta alone comprised on average about 11% of the population, compared to the corresponding 1-2% in France or England.
3. Because of Poland's republican style of government, public speaking, oratory and debate were essential for political participation. This was all carried out in Latin. Sarmatian culture also saw the Res Publica Poloniae as a "spiritual successor" of Rome and saw the Latin language as part of its identity. Furthermore, during the era of the elected monarchs, kings were not always fluent or able to speak in Polish, but they would have known Latin.