It was the norm for expeditions like this to go with a group of ships, and three seems to have been a common number. E.g. Bartolemew Dias' expedition of 1488 (first to reach cape of good hope) and Vasco da Gama's 1498 expedition (first to round cape of good hope and reach india) also used three ships. It was common for ships to be wrecked or have other problems that necessitated their abandonment, especially in unfamiliar waters, as happened on da Gama's expedition for example. With three ships if this happens the crew and cargo can be redistributed on the other ships, so it adds a kind of redundancy to these very risky expeditions. With one ship you don't have this. Decades of prior experience of these kinds of expeditions would have taught navigators like Columbus of the wisdom of having multiple ships. An even more extreme example of why this is necessary is Magellan's global circumnavigation of 1519 - 1522. He embarked with 5 ships, and only one returned. What if he had left with only one?

In addition to the safety aspects, there are other considerations. With multiple ships the fleet commander can send one to explore a certain area while the other ships carry out other tasks. Also, the purpose of these voyages in the age of discovery was to open trade and diplomatic relations with foreign countries. Columbus for example thought that he was going on a diplomatic and trade mission to the 'Great Khan of China', with the aim of opening trade relations and forming an alliance against the Muslims (he thought that the Emperor of China would be sympathetic to Christians due to outdated and faulty information). More ships = more space for cargo, and makes a more impressive sight to anyone they encounter, as well as being a better bet if you have to fight.