What does the phrase pee like a race horse mean?
You have to go so bad that you run or "race" to the bathroom! Racehorses are also sometimes administered a diuretic, lasix, which increases their urination.
More historical information
Horses are very modest creatures and most often will not urinate in public. Because the ones in captivity (ie race and show horses) spend so much time in their stalls, they get used to being there and feeling at home there. It often gets to the point where they will not leave their stalls to relieve themselves.
Because of this, they say that racehorses outside of their stalls are often in dire need of relieving themselves and are just waiting to be returned to the stall in which to do so.
An alternative explanation is that some racehorces are "bleeders". When bleeders race, they develop bleeding in their lungs which shows up as bleeding from the nostrils. Therefore, bleeders are administered lasix, a powerful diuretic, to induce increased urination. The ensuing volume depletion cuts down the bleeding and performance is improved. Thus, they "pee like a racehorce" from the lasix. Lasix is outlawed at some race tracks.