The strongest event that led to European colonization of the New World was the fall of Constantinople in 1453 to the Ottoman Turks. This meant that the way to Eastern silks and spices was closed to the Christian nations of Western Europe. This led Portugal and Spain to seriously seek a water route to Asia. North America was found by accident and soon Europeans sought to colonize it, always hoping that there would be a water route to the rich Asian markets.
There were other events and developments. Improved ships that could be trusted an long ocean voyages made the trip to the New World possible. Rivalries between England, France, and Spain made these countries feel pressure to gain the raw materials such as timber and gold from the New World with which to make war on their enemies. Gutenberg's printing press helped drive feelings of nationalism in Europe that made states powerful enough to organize these massive exploration attempts. Travelogues put before a public hungry for news also led other men to want to explore in order to get their names into the history books. The Protestant Reformation also spurred on European exploration as both Protestants and Catholics sought to recruit members for their side in the colonies. The discovery of North America also tied in with the Protestant way of thinking at the time that the discovery of a potential Eden meant that Christ's return must be close. There were many events that led to the discovery of America, but for me the single-most important one is the sack of Constantinople since that was the major driving force in improving maritime exploration in Western Europe.