The search for spices (or, more accurately, the search for ways to get to places that had spices) led to European exploration in the 1400s because it gave some Europeans a motive to go exploring. Once these Europeans had “discovered” a way around Africa and, more importantly, the New World, even more Europeans were inspired to explore in order to compete with them.
In the 1400s, the spice trade was dominated by the Venetians. They sailed their ships across the Mediterranean to places like Egypt where they bought spices that had been brought by Muslim traders from places like India. This was very lucrative for the Venetians.
During this time, the Portuguese had started sailing down the coast of Africa to trade. The domination of the spice trade by the Venetians, in addition to their growing skill as sailors, led the Portuguese to try hard to get to India by sea. They wanted to get some of the money that the Venetians were making. They eventually made their way around Africa and started getting spices on their own. Once the Spanish had finished expelling the Moors from Iberia, they too were interested in finding routes to India and they funded Columbus in his quest to get to India by sailing to the west.
Thus, the desire to take part in the spice trade led the Spanish and Portuguese to start exploring. Their “finds” led others, like the Dutch, the English, and the French, to explore as well.