'Jump the gun' derives from track and field races and was preceded in the USA by the phrase 'beat the gun' (or pistol). The earliest citation for the figurative usage, as we use the phrase today is from The Iowa Homestead, November 1921: "Give the pigs a good start; jump the gun, so to speak, and get them on a grain ration before weaning time." The figurative meaning is that something is started before the preparations for it have been completed.
"We realized that the important thing was not the film itself but that which the film provoked." Fernando Solanas