It's unclear exactly how this movement began, but much of the movement's public presence today is due in part to the work of filmmaker Lina Esco, who created a film titled, of course, Free the Nipple.
The general plot of the film, as according to the director herself, is about a revolutionary girl and the journalist who follows her during her topless protests. After watching her subject get arrested during one of her protests, the journalist (played by Esco) educates herself on the laws of NYC, and decides to figure out why we allow things like violence and swearing in the media—but not the female anatomy. The film culminates in a scene whereby several women run around in one of the most populated areas in the world, Times Square. Considering Esco started writing the film in 2012, it's been a major talking point in the movement, even before the film actually debuted in 2014.
If you want to go back much further, In Rochester, NY in 1986, seven women tried to do what guys had been doing legally for years—walk around topless. They were arrested in violation of New York state law. They were inevitably acquitted in 1992 by the state's supreme court, making N.Y. a legally topfree state for men and women. The case in Rochester was the legal starting ground for topless equality between men and women.