Originally Posted by garce

Compound prepositions are generally regarded as being informal. Tell him to pay more attention in class.

Otherwise, when a mechanical device is activated, it is "set off", or sent on its way. This relates to an object being put in motion. When your alarm clock sounds, it is effectively being sent into action: As per its mechanical predecessor, the act of "alarming" sets the device off and onto a course of action. In this case "off" does not mean silent, dormant, or motionless. It means that the device has been set off in an appropriate direction. Directed at getting your lazy ass off the comfy bed.