The most common explanation of why "cats and dogs" is used to describe a heavy rain is when, in olden times, homes had thatched roofs in which domestic animals like cats and dogs would like to hide. In heavy rain, the animals would either be washed out of the thatch or rapidly abandon it for better shelter, so it would seem to be raining cats and dogs. Other suggestions include derivation from a similar sounding, but unspecified, Greek aphorism which meant "an unlikely occurrence," or it is a corrupted version of a rare French word, catadoupe, meaning a waterfall. It has also been suggested, at one time, the streets of British towns were so poorly constructed that many cats and dogs would drown whenever there was a storm; people seeing the corpses floating by would think they had fallen from the sky, like the proverbial rains of frogs.

The most favored one in the references I have found is mythological. It seems cats were at one time thought to have influence over storms, especially by sailors, and dogs were symbols of storms, often accompanying images and descriptions of the Norse storm god Odin. So when some particularly violent tempest appeared, people suggested it was caused by cats (bringing the rain) and dogs (the wind).

There is, I have to report, no evidence I can find for any connection between the saying and the mythology other than the flat assertions of writers. The phrase first appears in its modern form in Jonathan Swift’s A Complete Collection of Polite and Ingenious Conversation in 1738: "I know Sir John will go, though he was sure it would rain cats and dogs," though a variant form is recorded in 1653 in City Wit, a work of the English playwright Richard Brome, in which he wrote "It shall raine ... Dogs and Polecats," which seems to suggest a stranger and less easily comprehensible origin.

There are other similes which employ falls of improbable objects as figurative ways of expressing the sensory overload of noise and confusion that can occur during a violent rainstorm; people have said it’s raining like pitchforks (first recorded in 1815), hammer handles and even chicken coops. It may be the version with cats and dogs fits into this model, without needing to invoke supernatural beliefs or inadequate drainage.

Having said all that, there is some evidence suggesting a direct link between heavy rain that seems to precipitate cats and dogs. It comes from a poem by Jonathan Swift, A Description of a City Shower:

Sweeping from butchers’ stalls, dung, guts, and blood;
Drown’d puppies, stinking sprats, all drench’d in mud,
Dead cats, and turnip-tops, come tumbling down the flood.

As Swift penned these lines in 1710, nearly 30 years before he wrote the book in which raining cats and dogs appears for the first time, it just might suggest he was quoting an expression he himself had created.

From World Wide Words http://www.worldwidewords.org/qa/qa-rai1.htm