Slogan for Kodak Photograph Company, invented by George Eastman himself, after his own amazement at the simplicity of the Kodak camera he had ordered to be made.
In 1888, professional photographers took the bulk of pictures in America. Most cameras were too cumbersome, expensive, and time-intensive for any but the most dedicated of amateurs. The Kodak changed all that, offering a simple shutter that could be opened and shut by clicking a single button.
Once the 100 pictures enclosed in every Kodak were used up, you simply sent the whole camera back to Eastman's Rochester factory, where they would develop the film, reload your camera, and return both to you for a nominal fee.
Hence the phrase: "You push the button, we do the rest."