Remember that labels for historical "ages" are imposed by later historians, and are to some extent arbitrary: people living in the Middle Ages, wouldn't really consider themselves in the "middle" of anything.

But the traditional account for the beginning of the Renaissance are that it's due to a number of significant events that happened around the 15th century:

1439 - The invention of the moveable type printing press in Europe, which increased the efficiency and effectiveness of spreading and retaining written knowledge.

1453 - The fall of Byzantium, bringing Greek scholars and their classical texts into Western Europe.

1492 - Columbus' discovery of America, providing a new continent for Europeans to explore and exploit

First, you have to remember that naming and labeling is something that is done by others, people living in the 13th century weren't sitting around discussing how they live in the Middle Ages.

The term ‘Middle Ages’ is the work of Italian historians during the renaissance, such as Flavio Biondo, who recognized the Middle Ages as a historical period distinctively different from their own during the fifteenth and sixteenth centuries. Conrad Rudolph contends that this was something Biondo’s contemporaries and immediate followers gradually formalized with terms such as media tempestas (1469), media aetas (1518), and media tempora (1531). Medium aevum is first found by at least 1604; with the English equivalent following immediately afterwards and is what we currently use.

Source:

Conrad Rudolph, A Companion to Medieval Art: Romanesque and Gothic in Northern Europe

I've always thought that around the Enlightenment period is when the Middle Ages ended. Though, I could be wrong and feel free to let me know if I am indeed wrong.