During the U.S. Prohibition, how did they get away with drinking alcohol?

A high demand for alcohol provided an economic incentive to those willing to break or circumvent the laws. The only problem was somehow breaking or circumventing the laws. The wikipedia article on alcohol during and after prohibition cites portable stills as going on sale within a week after prohibition went into effect and California grape growers as increasing production by a factor of seven as means to circumvent the laws.

Circumventing the law didn't satisfy the huge taste for alcoholic beverages. Most Americans who drank during Prohibition didn't drink homebrew. They instead drank illegal brew, illegal wine, and illegal spirits. They broke the laws by buying those illegal beverages. Others broke the laws by manufacturing, smuggling, transporting, and selling those illegal beverages.

What were smallish criminal gangs stepped in to fill that void. Those gangs made lots of money, and in the process changed from being small local gangs to huge organized criminal organizations. This meant they had to pay lots of bribes to keep law enforcement at bay, and even larger bribes and "campaign contributions" to keep law enforcement's elected superiors at bay. Al Capone (perhaps the most notorious mobster in American history) and William Hale Thompson (perhaps the most corrupt official in American history) figured prominently in this regard, but their story repeated itself across the nation. Every large city had a good number of speakeasies, and keeping those speakeasies open required lots of bribes. Lots and lots of bribes.

They probably got away with it because it was not illegal to drink alcohol. In fact, the Prohibition outlawed only the "manufacture, sale, or transportation" of alcoholic drinks. No mention of consumption, which remained substantial (~50-80% of "normal"), was made in the Prohibition amendment.

After one year from the ratification of this article the manufacture, sale, or transportation of intoxicating liquors within, the importation thereof into, or the exportation thereof from the United States and all the territory subject to the jurisdiction thereof for beverage purposes is hereby prohibited.

_{- Section 1, Amendment XVIII of the United States Constitution}

In fact, the National Prohibition Act of 1919, which implemented the Prohibition, contained a ton of exceptions and loopholes. One example is making wine at home; although the law called it "nonintoxicating cider and fruit juices", it was wine ^{[see note below]}.

The penalties provided in this Act against the manufacture of liquor without a permit shall not apply to a person for manufacturing nonintoxicating cider and fruit juices exclusively for use in his home, but such cider and fruit juices shall not be sold or delivered except to persons having permits to manufacture vinegar.

_{- Section 29, National Prohibition Act of 1919}

Therefore, in practice, people could keep on drinking by:

1. Consuming stockpiles purchased pre-Prohibition
2. Drinking as a guest at someone else's place
3. Making wine at home (often with ready-to-ferment "juice bricks")
4. Purchasing drinks on a "medical prescription" à la modern medicinal marijuana
5. Buying alcohol from organised crimes, who grew rich smuggling them in from Canada

The Prohibition also did not apply to drinks with less than 0.5% alcohol. Many breweries produced such malt beverages in order to survive the 1920s. Though, it is up to you whether to call that beer...

Note: this isn't actually on topic (consumption != manufacturing), but some comments were questioning the legality of making wine or cider at home. When the government tried to enforce the law, accused homebrewers invariably claimed their wine was not "intoxicating". While obviously not true, courts and judges a number of orders. The result was that the prosecution had to prove the home-brewed wine in question was "intoxicating in fact", a nebulous and undefined term.

We are forced to the conclusion that Congress ... required that, before a person can be convicted under the act for manufacturing such vinegar and fruit juices, same must be proved by the government to be in fact intoxicating.

We therefore hold that in all such cases it is necessary to prove that such vinegar and fruit juices are in fact intoxicating before a conviction can be had.

_{- Isner v. United States, 8 F.2d 487 (4th Cir. 1925)}

A famous case was that of three term Maryland Representative John Hill, a bitter opponent of the Prohibition who decided to make some wine and cider and reported himself.

[H]e hung fruit on the fence, gathered it a short time later, pressed it and permitted it to ferment, and telephoned prohibition agents that 'his wine and cider were breaking the law.' The arrested him and he was tried before a Federal jury. The juice was found to contain twelve per cent alcohol but the jury tasted it and stated that it was not 'intoxicating in fact' and released Hill.

_{- Maryland, a Guide to the Old Line State, Maryland Writers' Project, University of Maryland, 1940}

Thus it was legal to homebrew cider or wine as long as it was "not intoxicating in fact". The burden of proving it otherwise was placed on the Bureau of Prohibition, who decided to simply ignore homebrewers. This effectively legalised homebrewing in general.