Wordorigins.org

To rain cats and dogs is to rain very heavily. The metaphor behind the phrase is that of a dog and a cat fighting, something noisy and violent. The metaphor of a storm resembling a dog and cat fight dates to the mid-17th century, although the phrase, as we know it today, doesn’t appear for another half century. Henry Vaughn’s Olor Iscanus of 1651 goes:

The Pedlars of our age have business yet,
And gladly would against the Fayr-day fit
Themselves with such a Roofe, that can secure
Their Wares from Dogs and Cats rain’d in showre