As jlawler says in his comment, what's really going on is that [s] has become [h] in certain positions. This is a pretty common type of sound change, which falls under the rubric of "debuccalization". Phonetically, what happens in this type of change is basically that the articulation becomes laxer, in that (in this case) the near-closure at the alveolar ridge which is needed to produce an [s] sound is no longer being made, and all that's left in a puff of air, i.e. [h].
That's the "how"; as for the "why", that's a more difficult question, as is usually the case in linguistics generally and with sound changes specifically. Seeing that this kind of change occurs fairly often in unrelated languages (see the Wiki page for a similar example from Ancient Greek), it seems to be simply an articulatory tendency that some languages show and others not -- why is anyone's guess. I'm no Romance historical linguist, but French influence doesn't seem impossible in this particular case: the French were colonizing the Caribbean at about the same time that they were losing syllable-final consonants, and if French [s]-loss went through an [h] stage, it's possible that that remained as a substrate influence in Caribbean Spanish. But you could imagine other language-contact scenarios, e.g. possible influence from indigenous languages that lacked an [s] sound. In any case, though, a language-contact explanation isn't really needed here, since this is a common enough type of change that it could have happened on its own without any outside influence.