Finally! A question that relates to my field of expertise.

First, let's start with the origin of the Doubleday myth. The years after the US Civil War saw a shift in American preferences in sport. The war had spread a style of baseball common in the New York/New Jersey area (referred to as "Knickerbocker rules" or "New York game"). This slowly replaced regional variations of the sport which all more-or-less resembled the British schoolyard game of Rounders.

Baseball began to supplant Cricket as the popular team sport in the US during the Civil War. A field could be prepared much faster and easier than a proper cricket pitch. The veterans of the war carried the game and it professionalized and began organizing into leagues soon after the war (starting with the Cincinnati Red Stockings in 1869). Cricket, however, remained an amateur sport, but remained quite popular in places such as Philadelphia until around the First World War.

It was at this intersection of popularity, at about 1900-1905 that the question of baseball's origins was seriously considered. Albert Spalding, owner of Spalding Sporting Goods organized a commission to discover the origins of baseball... or more accurately, establish "the American origins" of the sport. During their research they received a letter that identified Abner Doubleday as the inventor of the sport, who devised it out of whole cloth as an original game in 1839 at his family farm near Cooperstown, New York. As far as the Mills Commission and Albert Spalding were concerend, it was the smoking gun that proved everything they set out to prove. That was all they needed, case closed.

The origins of the sport are a bit less clear cut, and they depend on what you mean by baseball. Evidence of versions of the sport date back to before the American Revolution. Games of "base ball", "stick ball", "town ball", "base", or "bass-ball" are mostly preserved as prohibitions in local statutes. The earliest recorded game of the "Canadian" version of baseball was in 1838, a year before Doubleday allegedly invented his game.

The rules of the New York game were first published in 1845 by Alexander Cartwright with the first recorded game in 1846 in Hoboken, NJ... but was probably played in some form of stickball on the streets and sandlots for a generation.

Sources

Block, David (ed.). Baseball Before We Knew It. University of Nebraska Press; 2005

Melville, Tom. The Tented Field: A History of Cricket in America. Bowling Green State University Popular Press; 1998

Morris, Peter. Catcher: How The Man Behind The Plate Became an American Folk Hero. Ivan R. Dee Publishing; Chicago, 2009

Schwarz, Alan. The Numbers Game: Baseball's Lifelong Fascination With Statistics. St. Martin's Press; New York, 2004