It's an interesting phrase. Even though it says "go off", it means to turn on. So the light bulb will turn on, meaning that you will understand. In cartoons, the artist will often draw a light bulb over a character's head that turns on when they have an idea or a sudden understanding.

You can understand it as 'lighting up' (switching on). Or you can understand it as 'going off' in the sense of being activated - the way that an alarm clock or burglar alarm goes off when it is triggered by a certain stimulus. You're right that, even as a metaphor, the idea of a light bulb making a sound is a strange idea, but if you think of it being activated by a sudden understanding or realisation, then it makes sense.