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How on earth was this film shown on mainstream theaters? How did they get away with it? We're talking about porn here, not an erotic flick.

In the late 1960's there were many changes dealing with sexually explicit material both in the US and in Europe.
In the US, municipal and state movie censorship ceased and the MPAA took over the movie classification which allowed american movie audiences to view more racy material. A number of legal decisions also permitted possession and distribution of explicit sexual material (which was considered a form of "free speech") as long as "community standards" allowed it.
Around 1968 the Danish government allowed the legal production/distribution of pornography. Lasse Braun and a number of early porn pioneers began making porn loops as well as feature-length hc movies. These were even screened at the Cannes film festival in the early 1970's.
Major newspapers in the US like the New York Times in the early 1970's ran ads for porn movies. In the US the situation did not last too long, with newspapers subsequently refusing to run ads (for XXX rated material) and with the Mayor and other blue nose groups shut down sex shops and adult movie theaters.
DEEP THROAT itself was released in June 1972 in New York and was a phenomenal box office success. Many personalities and celebrities even openyly admitted having seen it. It run in New York for many weeks and months until it was shut down for a while(on grounds of "obscenity").
For a more detailed account on the phenomen that was DEEEP THROAT and the tsunami it created, you should get your hands on the well-made documentary INSIDE DEEP THROAT (2005)
http://www.imdb.com/title/tt0418753/

Thanks.

It would take a book to describe the cultural and legal situation that existed at the time in the USA and elsewhere.
Acting in pornographic movies was considered criminal and essentially prostitution. Filming a hardcore movie in New York and elsewhere had to be done on the sly. If the film crew/performers were caught, the equipment would be confiscated and they would be arrested/prosecuted/fined.
Exhibition of such explicit material was problematic as well. Many filmmakers and exhibitors were frequently arrested and accused of "obscenity".
The FBI was playing cat and mouse with the showing of DEEP THROAT as they were trying to figure out where the box-office money was going. No one has even been able to figure out what it earned in its initial run.
The fact that certain films like DEEP THROAT in the early 1970's were being exhibited in more upscale parts of the city helped immensely and allowed the general population (those that would not dare frequent such seedy grind house porn theaters) to go see it. Word of mouth and news media that wanted to interview Linda Lovelace made it a cultural event never seen before.