There are a number of ways that you could look at this question. Let us give a brief answer to each.
First, the question asks why Spaniards (or Europeans in general) would want to come to a place like the Caribbean. The typical answer given for this is “God, gold, and glory.” The Spaniards wanted to find things to make them rich. They wanted gold and spices and other valuable things. The Spaniards wanted to increase Spain’s power and glory by giving it an empire and they wanted the personal glory of “discovering” new places. Finally, they wanted to bring Christianity to people who did not know God.
Second, the question asks why the Spaniards, rather than some other country, came to the Caribbean. In part, this is due to geography. Spain is geographically well-positioned to be a maritime country. It is also positioned in such a way that its sailors were likely to explore the Atlantic rather than the Mediterranean. Second, Spain was just lucky. The Portuguese were at least as active in exploration, but it was Spain that funded Columbus’s expedition and therefore it was Spain that got the “right” to the Caribbean in the Treaty of Tordesillas.
Finally, the question asks why it was the 16th century when the Spaniards came. One reason for this is that maritime technology was not as good in previous centuries. The lack of technology would have made long ocean voyages much harder in the past. More important, perhaps, was the fact that the Spanish had finally expelled the Muslims from Spain by the late 15th century. That left them with the ability to use their resources for something other than fighting the Muslims.