The Renaissance led indirectly to the "discovery" of the Americas because it created an intellectual atmosphere that promoted exploration.
Before the Renaissance, there was relatively little interest in science. People generally attributed everything to God's will and were not interested in deeper investigation. With the Renaissance, this changed. People started to think in a more scientific way and they started to be more interested in understanding the natural world. This attitude of wanting to know things helped to develop navigational tools that were needed for long voyages. This attitude also led to a desire to explore so as to find out what the rest of the world was truly like. In these ways, the intellectual attitude promoted by the Renaissance helped lead to the exploration that led to the discovery of the Americas.
The Renaissance certainly laid the grounds for a sense of intellectual curiosity about the world, an interest which led to technology that would make setting sail on treacherous waters much more "doable" for European explorers. The construction of fleets of ships, the advancement of mapmaking, the founding of navigational schools by such talented figures as Prince Henry of Portugal, and the creation of lateen sails, compasses, and astrolabes all proved to have a huge influence on mankind's ability to travel and conquer. At the same time, scientific ideals which had previously been held were increasingly being challenged. For example, Hermannus Contracts estimated the circumference of the earth and Thomas Aquinas used his theological power to suggest that the earth was round and not flat. This led Europeans to the gradual conclusion that there was more "out there" than what they had originally believed. Thus, explorers began to set sail during the fifteenth century to quench their financial, scientific, and intellectual thirst. In the process of sailing boldly West, these adventurers wound up running into the Americas.
The above answer by pohnpei397 is correct--the advances that allowed ships to navigate the oceans were invented by people who had a desire to know and learn. I would have to add that there were even more people who saw the monetary profits that could be made by finding new trading partners in far away places. The discoveries of the Renaissance also enabled the creation of new goods for that trade. It was a combination of pure scientific discovery and desire to trade that led, indirectly, to the voyages to the western hemisphere. For Ferdinand and Isabella, the rulers of Spain who funded Columbus, the incentive was new trade routes. They had defeated the Moors--who were Muslim--and chased them from Spain. By doing so they effectively closed their option of trading with the Muslim world, and needed new markets. None of this would have been possible, however, without the invention of such navigational tools as the astrolabe.