Learn English – Origin and exact meaning of “taken to the cleaners”

Wikipedia actually has an article dedicated to this phrase. It says:

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".

Wiktionary adds:

• The most common variation is to "see a man about a horse".
• Almost any noun can be substituted as a way of giving the hearer a hint about one's purpose in departing.
• The inversion to "see a dog about a man" eliminates any lingering uncertainty about whether the hearer is being put off.
• A shorter variant is to "see a man".

As to the exact situation in which you would use this phrase, it suggests:

Used as an excuse for leaving without giving the real reason (especially if the reason is to go to the toilet, or to have a drink)

Back to Wikipedia again,

During Prohibition in the United States, the phrase was most commonly used in relation to the consumption or purchase of alcoholic beverages.

World Wide Words has additional info:

This has been a useful (and usefully vague) excuse for absenting oneself from company for about 150 years, though the real reason for slipping away has not always been the same. [...] From other references at the time [around 1866] there were three possibilities: 1) [the speaker] needed to visit the loo [...] 2) he was in urgent need of a restorative drink, presumed alcoholic; or 3) he had a similarly urgent need to visit his mistress.

Of these reasons [...] the second became the most common sense during the Prohibition period. Now that society’s conventions have shifted to the point where none of these reasons need cause much remark, the utility of the phrase is greatly diminished and it is most often used in a facetious sense, if at all.

I found an 1889 example of honky-tonk: some 35 years earlier than the OED's 1924 honky-tonk, and some five years before their 1894 honk-a-tonk.

The Fort Worth Daily Gazette (Fort Worth, Tex.), of January 24, 1889:

A petition to the council is being circulated for signatures, asking that the Honky Tonk theater on Main street be reopened.

I found a good definition in The Iola Register (Iola, Kan.), June 23, 1893:

When a particularly vicious and low grade theater opens up in an Oklaholma town they call it a "honky-tonk." The name didn't just "come from" anything; it just growed.

The Sun (New York [N.Y.]), November 26, 1897:

BURNED DOWN THE "HONKY TONK"

Louisiana "regulators" break up a Vicious Resort and Shoot a Man.

NEW ORLEANS, Nov. 25.-Last night a party of regulators, about thirty-five in number, appeared at the Gramercy sugar refinery, in St. James parish, to break up a "Honky Tonk" there, where gambling, drinking, and all manner of vice prevailed. The regulators severely whipped the eight negro women in the "Honky Tonk" as well as the men they found there. Some of the negroes ran under the house to escape the beating. The building was set on fire and burned to the ground. Fears are entertained that some of the negroes were burned to death under it.

Oscar Dressner, a white storekeeper, who lived near the "Honky Tonk," came out to see what the regulators were doing, and they, fearing that he would recognize them, opened fire on him. He received four dangerous wounds in the back. Ho was brought to the Turo Infirmiry in New Orleans for treatment. He says he can Identify five of the men engaged In the affair.

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