To see a man about a dog or horse is an English idiom, usually used as a way to say one needs to apologise for one's imminent departure or absence—generally to euphemistically conceal one's true purpose, such as going to use the toilet or going to buy a drink.
The original, non-facetious meaning was probably to place or settle a bet on a racing dog.
Historical usage
The earliest confirmed publication is the 1866 Dion Boucicault play Flying Scud in which a character knowingly breezes past a difficult situation saying, "Excuse me Mr. Quail, I can't stop; I've got to see a man about a dog." In a listing for a 1939 revival on the NBC Radio program America's Lost Plays, Time magazine observed that the phrase was the play's "claim to fame".
During Prohibition in the United States, the phrase was most commonly used in relation to the consumption or purchase of alcoholic beverages.