how did philosophers of the middle ages study nature

The Middle Ages consisted of a period of almost a thousand years, from the fall of the Roman Empire to the start of the Renaissance. The Latin West consisted of all of Europe and Great Britain, and there were significant amounts of cultural interchange between the Latin West and Byzantium to the East and the Islamic empires to the South and East. This means that "philosophers of the Middle Ages" spans a very diverse group of philosophers and scientists of Europe.
The study of nature was usually called "natural philosophy." It was not a branch of empirical science, but rather an attempt to understand the fundamental nature of the universe. During the earlier part of the Middle Ages, it tended to be Platonic and focused on cosmology—especially Plato's Timaeus—and reconciliation of that with Christianity.
As more Greek works were recovered through encounters with Islamic writers such as Averroes and Byzantine refugees, natural philosophy increasingly focused on the study of Aristotle's works about nature.
A major impetus to the study of nature was the belief that God had composed "two books," the "Liber" (Bible) and the "Liber Mundi" (the book that is the world). By study of these two books, Christians could understand the mind of God. Understanding nature would also help readers interpret descriptions of elements of the natural the world in the Bible.
Toward the end of the Middle Ages, some scholars argued that the Liber Mundi represented the will of God more directly than the Liber and thus that one should observe the natural world directly to understand divine will. Medieval bestiaries, for example, combining the realistic and fantastical, interpreted animal behavior as moral and religious lessons.