Welcome aBoard, mila! Here are a couple of "explanations". I am not vouching for their authenticity. From Word for Word Archive:
Raining cats and dogs

Denny: Does anyone know the origin of the expression: "raining cats and dogs"?

Doug: This one harks back to ancient Germanic mythology. The cats are a symbol of the rain and the dogs symbolise the wind. So therefore any storm with a lot of heavy rain and strong gusting wind is said to be raining cats and dogs.

Dan: It started in the Middle Ages in Europe when the plague hit. People were so busy burying people they let the cats and dogs lay. When the next heavy rain came, the dead animals would come washing down the street. Hence the expression "raining cats and dogs".

Here's another, from Wordorigins.org:
Rain Cats and Dogs
Like many other phrases contained in these pages, the origin of this phrase is unknown. Its first recorded use is by Jonathan Swift in Polite Conversation, written circa 1708 and published thirty years later. This work of Swift's is a satire on the use of clichés, so the phrase was probably in use for a considerable period before this. Rawson reports an earlier variant, "rain dogs and polecats," from Richard Brome's The City Witt of 1652.
Suggestions for its origin include:

The archaic French catdoupe, meaning waterfall or cataract.
The thunder and lightning are akin to the sound of a cat-and-dog fight.
In Norse mythology, cats had an influence on the weather, and Odin, the sky god, was attended to by wolves.
All of these are unlikely given the Brome cite. Since the earliest English language variation starts with "polecat" as opposed to "cat," the idea that it descends from catdoupe is unlikely. Similarly, since polecats are not cats, but rather weasels or skunks, the connotation with mythological properties of felines is similarly unlikely.
Yet another explanation is that in old London a hard rain would fill the streets with water and drown many stray dogs and cats. Thus when the rain stopped and the water was gone, it looked as if it had rained cats and dogs. It sounds plausible, but the early use of the phrase with polecats make it unlikely.

So, the mystery remains with no good explanations.