Your brain is basically a toddler running on no sleep, and it has opinions about everything, including the weather.
When you're tired, a light drizzle outside doesn't just sound like rain. It sounds like the sky is personally attacking your roof with a fire hose. Scientists will tell you it's something to do with acoustics and perception. But the real answer is simpler and a little funnier than that. Your brain, when underpowered, turns up the volume on absolutely everything. The heating clicks. The fridge hums. Your neighbour's dog sneezes three houses down and suddenly it sounds like a foghorn.