No doubt you will find that different people produce a different answer to this question. My research indicates that the idiom “raining cats and dogs” probably comes from a poem by Jonathan Swift entitled A Description of a City Shower. The poem was published in 1814 in The Works of Jonathan Swift: Miscellaneous Poems by Jonathan Swift, Sir Walter Scott.
The poem apparently dates to 1710. It describes London, England. The poem appears to have been published in the Tatler. As you can see the last two lines give us a clue as to where the origin of this saying comes from.
However, those lies do not explain to us their true origin. It appears that in 1653 a playwright Richard Brome wrote a comedy The City Wit All the Woman Wears the Breeches in which he mentioned stormy weather, “it shall raine… Dogs and Polecats”.
It seems that you can go further back in time for the origin of this idiom. A book published in 1592 by Gabriel Harvey called Pierce’s Supererogation:
In steed of thunderboltese, shooteth nothing but dogboltes, or catboltes.
Of course, the language looks peculiar by today’s standards. But you can also clearly see the reference to the current saying. There is reference to “thunderbolts” i.e.thunder and lighting accompanied by heavy rain. However, having done this research, I am still unsure of the reason why cats and dogs are used to emphasise the heaviness of the rain! Perhaps the simple explanation is that cats and dogs are much larger than raindrops and therefore when it is raining heavily the raindrops are fictionally as big as cats and dogs. Does somebody know any better than me?