In May of 1498, da Gama and his men reached India. It is said that an Arab merchant who knew how to speak Spanish asked them what had brought them to India. They are said to have replied “Christians and spices.” This is, in essence, what motivated the voyages of da Gama.
Part of the reason why da Gama and other Europeans explored was a desire to spread the Christian faith. This was a much more universally religious time and Christians tended to believe that anyone who did not share their faith was going to be damned for all eternity. They felt it important to try to save others from that fate.
The other half of the equation, though, was economic. Europeans needed spices and the only source for these spices was Asia. The spices came to Europe via the Middle East and Italy. This meant that Muslims and Italians controlled the spice trade. Portugal and Spain, in particular, wanted to trade directly with the areas that had the spices so they (instead of the Muslims and Italians) could profit. There was a tremendous amount of money to be made. That is much of why da Gama explored (and why Columbus sailed as well).
So, “Christians and spices” is a pretty good way to sum up the motivations for da Gama’s exploration.
Portugal wanted to find a sea route to India and the Far East in order to engage in the lucrative spice trade, especially in cinnamon and pepper, without going through middlemen. The land route across the Arabia, dominated by Muslims, was dangerous, as was sailing the Mediterranean Sea, where a boat could meet with pirates or ships from other countries disputing Portugal's right to be there. For years, under Henry the Navigator, the Portuguese had been exploring the idea of an ocean route, and in 1487, before Vasco da Gama travelled, had discovered that the Atlantic and Indian oceans met at the southern tip of Africa, meaning it was possible to get to India by sailing around Africa.
The success of da Gama's mission meant that Portugal prospered economically for a time. Portugal's king, Manuel, also hoped to use a new sea route to defeat the Muslims and retake Jerusalem for the Christians, making himself king of Jerusalem, but these religious visions went unfulfilled.