“To rain cats and dogs” means to rain heavily. The idiom has been cited in print since at least the 17th century.
 
“It was raining cats and dogs and I stepped in a poodle” (a pun for “puddle”) is an old joke that he been frequently included in joke collections. American author and newspaper columnist Lucius Beebe (1902-1966) said in 1932:
 
“It really is raining cats and dogs. I know. I just stepped in a poodle.”
 
It’s not known if Beebe coined the joke or if the joke has an earlier source.
 
“What’s worse than raining cats and dogs? Hailing taxis” is a related joke.
 
 
(Oxford English Dictionary)
to rain cats and dogs: to rain very heavily. Also attrib., raining heavily. Similarly to blow (also pour) cats and dogs .
[a1652   R. Brome City Wit iv. i. sig. D7v, in Five New Playes (1653) ,  It shall raine..Dogs and Polecats.]
1661   T. Flatman Don Juan Lamberto (new ed.) ii. iv. sig. L,  Here we have Houses over our heads, so that if it should rain Dogs and Cats we could have no harm.
1738   Swift Compl. Coll. Genteel Conversat. 178,  I know Sir John will go, tho’ he was sure it would rain Cats and Dogs.
1766   P. Thicknesse Observ. Customs Fr. Nation 106   It blows cats and dogs, as the sailors say.
 
25 September 1932, Richmond (VA) Times-Dispatch, pg. 9, col. 8:
“He (American author Lucius Beebe, 1902-1966.—ed.) said, ‘It really is raining cats and dogs. I know. I just stepped in a poodle.’”
 
17 April 1934, Chicago Heights (Chicago, IL), Star,  “Pete’s Pet Peppers” by Pete from Crete, pg. 9, col. 4:
Absolutely the worst joke we ever heard was the one we heard this way:
Cowdery Miller—It’s raining cats and dogs outside.
Oliver Glawe—How do you know.
Cowdery—I just stepped in a mutt poodle.
 
Google News Archive
4 September 1936, The Cayuga Chief (Weedsport, NY), “Will You Smile?”, pg. 8, col. 2:
Tsk! Tsk!
Sailor — It’s certainly raining cats and dogs, isn’t it?
Girl Friend — I just stepped in a poodle.
 
Google Books
Isaac Asimov’s Treasury of Humor:
A Lifetime Collection of Favorite Jokes, Anecdotes, and Limericks with Copious Notes on How to Tell Them and Why
By Isaac Asimov
Boston, MA: Houghton Mifflin
1971
Pg. 167:
His sympathetic wife said, “Oh dear, it’s raining cats and dogs outside.”
“You’re telling me,” said Smith. “I just stepped in a poodle.”
 
Google Books
Martha Speaks:
Funny Bone Jokes and Riddles
By Susan Meddaugh
New York, NY: Houghton Mifflin Harcourt Publishing Company
2013
Pg. 7:
Why do you have to be careful when it rains cats and dogs?
You might step on a poodle