There were a few factors that led to Columbus' exploration. The Portuguese were the first to find a new trade route to India when Vasco da Gama successfully navigated around the Cape of Good Hope. This voyage opened up a direct link to the spice trade in India/Asia and Europe. Columbus, motivated by what is commonly referred to as the 3G's, wanted to find a route to Asia in order to find gold, achieve glory, and as a secondary mission, spread religion (God). He had the idea that if he were to sail west, which no European had done before*, he could find a faster route to China and the Spice Islands (Malaysia and Indonesia today). After having his plans rejected by several monarchs, eventually Queen Isabella and King Ferdinand of Spain agreed to fund three ships for Columbus to sail west in order to find another route to Asia.
Among the motivations for Columbus' voyage were the 3 Gs, as discussed above. Prior to sailing, the Europeans traded with the East on the Silk Roads. Land-based routes passed through the Byzantine City of Constantinople. When Constantinople fell to the Seljuk Turks in 1453, it became Istanbul, an Islamic city. This helped Muslim merchants take more control over trade routes between Europe and Asia. This motivated the Europeans to find new trade routes in two ways: first, the Portuguese and Spanish governments maintained residual intolerance for Muslims and Islam in general which was left over from the Crusades (a religious war between Christians and Muslims over land and religious sites), and second the Portuguese and Spanish were interested in having a direct route to Asia in order to cut costs. If you consider the distance some goods traveled on the Silk Roads and over the Indian Ocean, and the amount of times those goods changed hands at cities and trading ports along the way, goods from Asia were very expensive for Europeans. If the Portuguese and Spanish were able to find a direct route to Asia, they would not have to pay handling fees and could make more profit from trading Asian goods.
Wanting to find a new route to Asia, and motivated by the prospect of glory and gold, Columbus set out in 1492 to find a western route to the spice trade. As we know today, he actually landed in the Bahamas, and his journey sparked interest in what would later be confirmed as the "New World", and not the Eastern edge of Asia.
*The Vikings sailed across the Atlantic and settled in modern-day Canada, so Columbus was not the first to make the voyage. However, he was the first Age of Exploration explorer to make the voyage.