"Because the paper beds of banknote presses in 1860 were 14.5 inches by 16.5 inches, a movie industry cartel set a standard for theater projectors based on silent film, and two kilobytes is two kilobytes" is as far back as I have been able to push this, but let's get started.
In August of 1861, by order of the U.S. Congress and in order to fund the Union's ongoing war efforts against the treasonous secessionists of the South, the American Banknote Company started printing what were then called "Demand Notes", but soon widely known as "greenbacks".
It's difficult to research anything about the early days of American currency on Wikipedia these days; that space has been thoroughly colonized by the goldbug/sovcit cranks. You wouldn't notice it from a casual examination, which is of course the plan; that festering rathole is tucked away down in the references, where articles will fold a seemingly innocuous line somewhere into the middle, tagged with an exceptionally dodgy reference. You'll learn that "the shift from demand notes to treasury notes meant they could no longer be redeemed for gold coins" -- which is strictly true! -- but if you chase down that footnote you wind up somewhere with a name like "Lincoln's Treason -- Fiat Currency, Maritime Law And The U.S. Treasury's Conspiracy To Enslave America", which I promise I am only barely exaggerating about.
An awesome historical investigation of why terminals were 80×25: