How did discoveries in science lead to a new way of thinking for Europeans?
First, it should be noted that while older histories of science and philosophy, written on a fairly elementary level, emphasize discontinuities, more recent scholarship and scholarly monographs tend to see the development of science as continuous.
The first great period of scientific innovation in Europe was the so-called "twelfth century Renaissance". Europeans rediscovered earlier Greek science and math through the Crusades, which brought them into contact with Islamic and Byzantine cultures. Among the most important developments of the period was the importing of algebra from the Islamic world to Europe. This affects science as what distinguishes much of modern science from other fields is its use of mathematics to represent observed regularities in the world.
The next change scientific discoveries brought to European thinking was a sort of technological optimism, a belief that the mysteries and problems of the world could be discovered and solved by means of observation and experimentation.
Another change was religious. The notion that the material world followed regular rules which could be expressed mathematically supported the notion of the "watchmaker" God who set the universe in motion according to fixed laws.
Discoveries in science led to a new way of thinking for Europeans because they called into question various ideas that had been taken on the authority of the Bible or the authority of classical thinkers like Aristotle.
As the Scientific Revolution was getting started, people like Copernicus and Kepler started to discover that the geocentric model of the universe was wrong. Kepler, in particular, used observations (rather then theorizing) to confirm a heliocentric model of the universe. Similarly, William Harvey used experiments and observation to prove that the heart was the beginning of the circulatory system. This disproved classical ideas which said the liver played this role.
These sorts of things led to a whole new way of thinking in which hypothesis and experiment and observation (as opposed to relying on past authorities) were the path to knowledge.