What does piss like a Russian race horse mean?

Answer 1

All this silly phrase means is to urinate in large quantities, as any horse naturally would. The Russian racehorse part is just America's love of alliteration{ we like words that start with the same letter used one after another].

Answer 2

It means you have to pee really bad, and the phrase is 'like a rushing race horse' lol.

My understanding from what my grandfather told me when I was very young is the following:

The Kentucky Derby was aired only on the radio before the advent of television, obviously. But once it became a television event, they would introduce the racehorses, showing them on camera, and tell us what country the horses originated from. And it seemed that everytime they showed the Russian horses on camera, they were always peeing, or at least more so than the horses from other countries. So some famous comedian in the 1950's (I can't remember who my grandfather said it was) made a joke about it during one of his comedy routines, and the phrase was born.