Rainy weather becomes depressing because of the lack of sunlight.
When the sun is shining we get Vitamins from the sun but when it's cloudy out those Vitamins are lacking.
The rainy weather can also become depressing because of not being able to get outside and being stuck inside with nothing else too do.
Serotonin which is a hormone that plays an important role in regulating our mood is lacking due to the lack of sunlight as well.
When our Serotonin levels are lower then our mood gets into depression mode.