The Renaissance did this through things like art, literature, and later through science.
Looking at art, we can see that during the Renaissance artists came to cultivate their own styles. During the Middle Ages, art had tended to be fairly uniform, with the emphasis on the religious subject of the art. In the Renaissance, art continued to be religious in many cases, but it started to look very carefully at the human individual (think of how individual and well-portrayed the people in Michelangelo's paintings are) and it started to be done in individualistic styles. This sort of thing would have encouraged people to think of themselves in more individualistic terms. So would the writings of people like Petrarch and Dante. These ideas were then taught in the new schools that focused on humanism in education. Eventually, they led to the scientific revolution that really brought individuals' abilities to think and reason to the fore.