How and why did the indigenous people arrive in the Caribbean?

Let me first address the "why" part of your question, as the "how" will be a little more complex. Most human migration, especially prior to the Modern period, was driven by a need or desire for resources. Even well into the Modern period, First Nations people (like the indigenous Taino and Arawak) lived highly mobile lifestyles driven by access to seasonal resources. So, why did indigenous Caribbeans migrate there? For access to food and other resources!
As I mentioned, the "how" is a little more tricky. We do not have historical records detailing the peopling of the Americas and must rely on archaeological materials and "best guesses." Archaeological evidence indicates that people were living in Trinidad at least eight thousand years ago, and by seven thousand years ago were continually pushing north towards Cuba. Based on this evidence, we could infer that once groups of people had arrived in South America, they began pushing northward into the islands of the Caribbean. There is also evidence to suggest that people migrated south from present-day Florida or east from the Yucatan peninsula.
How did these people get to the Caribbean islands? Based on the sea-floor geography and fluctuations in sea levels, they might've been able to walk or wade easily from one area to another! It is also possible that these people were building rafts or boats and using them to cross longer distances. It is rather unlikely that the peopling of the Caribbean was accomplished entirely by swimming, even though it is possible to swim for long distances. Children, the elderly, and the infirm would not have been able to make such long swims, so the use of a boat or raft would have been needed.
https://www.archaeology.org/news/3504-150722-americas-genetic-migration

https://www.britannica.com/place/Caribbean-Sea

https://www.culturalsurvival.org/publications/cultural-survival-quarterly/indigenous-people-caribbean