This superstition is based on ancient Japanese wisdom when people took advantage of the meteorology mastered by cats, and it has since been supported by science.
Cats can sense moisture on their whiskers and feel uncomfortable when that moisture increases due to humidity. So they wipe their faces with their paw, which looks like they are washing their faces.
Whiskers are pretty much the same as fur hair, only thicker and stiffer. They're inert and contain no nerves, so it's impossible for cats to "sense moisture on their whiskers". It's the follicles of whiskers that have the nerves, but their job is to sense the movement and/or vibration of the whiskers, not ambient moisture.
A cat's head comprises vital organs such as the mouth and nose for smelling, tasting, breathing, eating and drinking, eyes for seeing, and ears for hearing and balancing. As with the human head, these organs are particularly important and the cat knows how to look after them.
A cat washes its face for several reasons, including; to regulate body temperature through evaporation of saliva, stimulate circulation, keep its coat clean and smooth by distributing natural skin oils, treat infection, eliminate parasites, or simply because they like doing it.
Especially after eating or foraging, the cat washes away any bits of food or other smelly material from its face fur, whiskers and skin to avoid contaminating odours of other food. The cat's sense of smell is important for deciding whether something might be edible.
Cats clean themselves frequently; somewhere between 30% and 50% of the time they are awake. So if the superstition were true then it would be raining a lot more than it does.
Incidentally, cats clean their faces just as frequently in the driest places on Earth which have hardly any rain. Conversley, it rains a lot over the seas and oceans where there are no cats at all.