As everyone knows, the saying is used to define a particularly heavy rain. Italian people say "Piove a catinelle", "It's raining basins", far less imaginative.
It seems that there is no sure explanation, but the most likely one might be linked to the fact that in 1500, roofs were made of simple straw, piled up in thick heaps and bonded together
Some animals such as dogs, cats and rats, loved to rest on those soft "cushions", but it often happened that the rain made the straw slippery and that the poor animals would fall or literally "rain" from the roofs.
Some people prefer to believe instead that this saying was what some very creative mind came up with in order to highlight in a figurative way how the rain is often dense and heavy here in the UK.
In some areas you can still see these old straw roofs (the Girlfriend has personally seen several of them) Maybe there "raining cats and dogs" is not only a metaphor.