Wordorigins.org

These are American slang terms for an illegal drinking establishment. Blind tiger is most commonly found in the South. Its synonym, blind pig, is more common in California, the Northwest, and the Northern Tier states.

The terms apparently arose from the fact that it was illegal in many areas to sell alcohol by the drink or because the term was used to avoid licensing fees. Proprietors of drinking establishments would advertise animal curiosities and give customers who paid to see the animals a “complimentary” drink. Often, there were no actual animals to be displayed and it was a thin facade that law enforcement officials winked at.

Blind tiger is first cited in May 1857, in the magazine Spirit of the Times:

I sees a kinder pigeon-hole cut in the side of a house, and over the hole, in big writin’, “Blind Tiger, ten cents a sight.”...That “blind tiger” was an arrangement to evade the law, which won’t let ‘em sell licker there, except by the gallon.

Blind pig appears a bit later. The Historical Dictionary of American Slang has this from 1886:

Mysterious beverages...from a blind pig in Iowa.

And Minnesota passed a law against blind pigs in 1887:

Whoever shall attempt to evade or violate any of the laws of this state...by means of the artifice or contrivance known as the “Blind Pig” or “Hole in the Wall”...shall...be punished.

(Sources: Oxford English Dictionary, 2nd Edition; Historical Dictionary of American Slang; Dictionary of American Regional English)