From their website:

Charles Feltman, a German immigrant, invented the hot dog in Coney Island during the summer of 1867, as a convenient way for beach goers to enjoy quality sausages on a bun without plates & silverware, by turning his pie cart into a hot dog cart. From those humble beginnings, Feltman’s became, not just a restaurant, but a symbol of American tradition & quality on Coney Island. Since Feltman’s closure in 1954, the hot dog has slowly become degraded to a food of poor quality with fillers and nitrates. Feltman’s of Coney Island is determined to bring back that tradition of quality to the great American Hot Dog!

Wikipedia, on the building above:

The land was later used to construct the Astroland amusement park which opened in 1962 and closed in 2008, subsequently replaced by a new Luna Park. The last remnant of Feltman's – the building that had housed the kitchen – was demolished in 2010.