How and why did Europe transition from the Medieval Period to the Renaissance?

Europe's transition from the Medieval Period to the Renaissance was set into motion by social and economic crises that eroded the foundations of medieval society in the 1300s and 1400s. The waves of plague that swept through Europe had, by the end of the 1300s, reduced the population by as much as a third and caused a scarcity of labor. This, in turn, provided leverage for the peasants to demand lower rents and better wages from the nobles who owned the feudal estates. With unmet demands came the peasant rebellions and the movement of populations from rural to urban environments. These changes began to unravel a feudal system that had for so long shaped the outlook of the European people. Commercial and manufacturing activities in towns, especially along trade routes, brought about the rise of a commercial or middle class (made up of artisans, merchants, and shopkeepers) that no longer relied heavily on the church for guidance. The freedom to work for and create prosperity for oneself, rather than for a feudal lord, introduced the value of individualism and set the stage for the Renaissance, or rebirth, of ideals from classical Greek and Roman culture, with a focus on learning, personal achievement, and refinement. Trade also played an important role, as many of the ancient classics were reintroduced into Western Europe by Europeans who came into contact with Byzantine and Arab scholars. This new era first took root in the prosperous northern Italian city-states and then spread through Europe between the 1300s and the mid-1600s.

This is a tricky question. Generally, these time periods are named after the fact and the differences in the eras of places like Europe were both gradual and natural. When we think of recent American history, we can think of eras like that of the Great Depression and point to specific events that lead to that happening; instead, when Europe transitioned from the Middle Ages to the Renaissance, there were many small factors and changes over time that led to a difference in society and culture at large.
But, with this acknowledged, let's look at the eras as we know them today. In general, we look at Western history as divided into antiquity, medieval period (or middle ages) and modern. The Renaissance can be thought of as the period of time in between the middle ages and the modern period, linking the two and causing the changes in Europe that really allowed it to transition between those two main periods. The Renaissance is generally understood to have begun in Italy around the end of the 13th century, when certain artists began to produce humanistic works and a new era of creativity and artistic production started to begin. This basically formed a new set of artistic and cultural values that carried on to the rest of Europe.
Of course, this theory is just that: a theory. Because these periods are rather vague and debated, it's hard to say something like "well, Petrarch did X, so Y happened and then it was the Renaissance." Another theory that has been proposed that I find interesting is that of the Black Death. Because everyone (like, really, nearly everyone) was dying because of the plague, it's possible that people stopped caring so much about things based less on human reality (think spiritualism, religion, etc.) and their interests and priorities started to shift more towards things here on Earth, namely humans and art created about them.
I hope this is somewhat helpful. There are a lot of great resources about the Renaissance period online, do a bit more research if you need more info! Good luck.