The factors that led to European exploration in the 15th century were multi-faceted. Initially, they were motivated by a desire to find faster and cheaper trade routes to Asia, where they could trade for spices. Spices were a lucrative commodity in Europe. For centuries, the spice trade was dominated by Muslim merchants who brought these spices to Europe through overland routes and charged a high premium for them. European merchants hoped that by establishing their own sea-routes, they could go directly to the source and gain better access to the spice trade. This was the primary motivation of Vasco de Gama and Christopher Columbus.
Another factor that aided European exploration during this period was improvements in sea-going technology. Better ship-building techniques and navigational advances, such as an improved compass and sextant, in combination with advances in cartography (map-making) meant that it became easier, faster, and relatively safer for European sailors to head out on long voyages.
There was a competitive aspect as well. For instance, Portugal and Spain were in a competition for access to better trade routes. They each figured that control of certain routes and foreign markets meant that they could get the upper hand over each other both economically and politically. A similar sense of competition existed between various Italian states. This sense of competition compelled many to set out on voyages of exploration.
Also, while it would become more of a motivation during the 16th century, the desire to spread Christianity was yet another reason for exploration. Most of the inhabitants of the lands beyond Europe belonged to other faiths, and there was a strong desire by many Europeans to evangelize in these new lands.