Why are droughts a problem on some Caribbean islands

Depends on the problem you're addressing.

Is it how are droughts a problem? Or why are droughts a problem?

To the later, I would argue that because most if not all water on and around caribean islands is salt water, and consequently the relliance on rain for fresh water is greater.

With fresh water you can cook, cultivate and drink. Where with saltwater you can't cook or cultivate land and drinking will eventually cause dillusions and death.

why are droughts a problem in ANY country? They can't grow their food. Granted, in a country such as the states, there is easy trade... you can just ship your food somewhere just as easily.

In the islands, since they are an island, it is harder to simply put food on a truck and drive a few miles.

Also, the Caribbean islands are known for their cash crops: sugar, coffee, cacao, coconuts, etc. No rain = no money.