It means to politely excuse yourself from a situation to go to the restroom or buy a drink. It originated from men disappearing to go bet on horse or dog races. See a man about a dog means the same thing.
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 is the play's "claim to fame".

During Prohibition in the United States, the phrase see a man about a horse was most commonly used in relation to the consumption or purchase of alcoholic beverages.
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Phrase used to conceal one's true destination, especially in response to an awkward question. Is commonly used in Britain when one is headed to the toilet. See also going to see a man about a dog
"Are you and your wife coming out tonight?" "Uhhhh, I need to see a man about a dog."

"How come you're leaving the table?" "I need to see a man about a dog."
by Alexandra July 29, 2004
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Any general business that needs attending to that you may not care to discuss whith the present party.
We can meet for lunch on Tuesday, since I'll be in the area; I have to see an man about a horse.
by moth17 July 5, 2005
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go to the bathroom.

While it may be used in "Finding Nemo", Mr. Disney (who was deceased, by the way, when "Finding Nemo" was made) did not come up with it. It is an Austrailian saying which was popularized in the movie "Crocodile Dundee".
"Man, that was a great chili dog, but now I've gotta see a man about a wallaby."
by ixolib December 3, 2006
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Phrase invented in the early 1900s that merely means that you have to use the restroom. Some also use it because they do not want to reveal the true nature of their business.
Where are you going?" "I'm going to see a man about a dog, I've been holding it all day!
by ZestyBacon January 28, 2011
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