A sound is raspy when it is rough or scratchy. If you get a bad cold or scream encouragement to a losing team for a few hours, you will end up with a raspy voice.
A rasp is a tool, usually used by wood- or metal-workers, to scrape. Nail files and kitchen graters do the same work that a rasp does. A raspy voice sounds like it had to pass through a rasp or a grater to get out of your mouth. If you can barely speak, you might ask for water in a raspy whisper. Other sounds can seem raspy too like a dry cough, the caw of a crow or the bark of a dog that has been at it for a long time.