Some even create discussion of the topic at BBC, and propose the answers like as follows: "Peasants used to live in tiny hovels with thatched straw roofs. Their cats and dogs would live outside and often climbed onto the roof to bed down for the night, presumably warmed by the heat from the fires inside the hovels. When there was very heavy rain falling, the straw would become very slippery and the animals often fell to the ground!" http://www.bbc.co.uk/bristol/content/weather/2003/02/28/raining.shtml
Imagine this... :)
The British scholars, in their turn, attribute this expression to J.Swift, 1710, as it is first found in a text by him. Yet, a similar phrase was recorded already in 1653 ("It shall raine... Dogs and Polecats").
It does not mean, however, that this expression did not exist before then. And very likely it derived from something closer to real rain, shower and storm:
1) "CAT" = SCAT (Eng., rare use) - a stormy rainfall. Comp. to: CADO (Lat.) - to fall. Compare also to: CATARACTA (Lat.), CATADOUPE (Fr.) - waterfall; KAATAA (Fin.) - to pour, to spill; KAATOSADE (Fin.) - torrential rain; KET (Komi), KOT (Udmurt) - wet; KHATAMIRA (Chewa language in South Africa) - to be wet with rain (body or ground), anything cooked with too much water.
2) "DOG" = DAG (Eng.) - thin rain, drizzle, wet fog; DÖGG (Icelandic), DAGG (Sw.), DUG (Dan.), DUGG (Norw.) - DEW (Eng.). Comp. also to: DUSK-REGN (Sw., Norw.) - DUSK-RAIN; DOCCIA (Ital.), DOUCHE (Fr.) - shower; ДЪЖГЬ [DUZHG'] (outdated Rus. Novg. and Pskov dialects), DESZCZ (Polish), ДОЖДЬ [DOZHD'] (Rus.), etc. - any rain; TUHU (Karel.), TUISKU (Fin.) - blizzard, windy snowstorm; DOGI (Ingush), DOG'A (Chechen) - rain; DOŠ (Ingush) - wet; TU, T'USK (Sumer.) - to bath, to wash; DUG.GA (Sum.) - to irrigate; DIG (Sum.) - to moist, to wet.
There are, of course, many more words in these two semantic fields, backdating to as far as the Sumerian epoche and texts, having left trace in the numerous language families worldwide.
Gary Martin, a British scholar, writes, however:
'...In truth, what was in the mind of whoever coined this expression is now lost to us. I have to admit defeat and say that I don't know the origin of this phrase.' 
British linguists seem to be confident that they have a broad enough written record of Old English to only take that record into account ?
Very likely also related: KOSK (Est.), KOSKI (Fin., Karelian), KOS'K (Vepssian) - a waterfall (sing.); KOSED (Est.), etc. - waterfalls (pl., with -D(-T) being the Finnic (and Celtic) plural ending).
CASCADE (Eng.), CASCADA (Sp.), KASKAD (Sw.), КАСКАД [KASKAD] (Rus.) - a waterfall, or a series of waterfalls (with -D already absorbed in the 'I.-E.' langs as if it were a part of the stem!); CASCADE-S, CASCADA-S, KASKAD-ER, КАСКАД-Ы - same words in plural, but with the 'I.-E.' plural endings added upon the already existing Finnic (or Celtic) plural ending.
 Simo Parpola (Eisenbrauns, 2016). Etymological Dictionary of the Sumerian Language.
 TUHU (Karel.) - snowstorm, e.g.: Tuhu kai dorogat umbai; nengomal tuhul putin ižändy ni koirua pihale ei työnä (Karel.)
– The snowstorm picked up all the roads; in such a snowstorm a good owner will not let out even a dog into the street / Метель все дороги замела; в такую метель хороший хозяин и собаку не выпустит на улицу.
In this proverb a real dog is mentioned.
Learn more: https://new-etymology.livejournal.com/14204.html