This is an interesting phrase in that, although there's no definitive origin, there are several speculative derivations.
The phrase seems isn't related to the well-known antipathy between dogs and cats, which is made word in the phrase 'fight like cat and dog'. Aside from the presence of cat and dog in the phrase, there's nothing at all to connect their fighting with raining.
Nor is the phrase in any sense literal, i.e. recording the fact that cats and dogs fell from the sky. Numbers of small creatures, of the size of frogs or fish, do occasionally get carried skywards in freak weather. That must happen to individual dogs or cats from time to time too, but there's no record of groups of them being scooped up in that way. Not that we'd need meteorological record for that - it's plainly implausible.
In fact, 'raining cats and dogs' only makes sense figuratively and the explanations below that attempt to link the phrase to felines, canines and weather seem rather feeble.
http://www.phrases.org.uk/meanings/raining%20c ats%20and%20dogs.html