How did the expression "hands down" come to mean "without a doubt?"
The phrase hands down comes from horse racing, as explained by The Phrase Finder:
Jockeys need to keep a tight rein in order to encourage their horse to run. Anyone who is so far ahead that he can afford to slacken off and still win he can drop his hands and loosen the reins - hence winning 'hands down'.
And confirmed by Etymonline:
To win something hands down (1867) is from horse racing, from a jockey's gesture of letting the reins go loose in an easy victory.
And I found examples of the phrase being used figuratively in other contests by the early 1880s.
Win easily, with little effort.
Jockeys need to keep a tight rein in order to encourage their horse to run. Anyone who is so far ahead that he can afford to slacken off and still win he can drop his hands and loosen the reins - hence winning 'hands down'. This is recorded from the mid 19th century. For example, 'Pips' Lyrics & Lays, 1867:
"There were good horses in those days, as he can well recall, But Barker upon Elepoo, hands down, shot by them all."
It began to be used in a figurative sense, to denote an easy win in other contexts, from the early 20th century.
It may very well be from horse racing, but "keeping a tight rein" is not to encourage a horse to run. A tight rein is restrictive. A horse is "given his head" in order for him to run freely. My suspicion, if horses are involved, is that "hands down" indicates that the jockey is relaxed, no longer in racing stance, having no need to further communicate with or guide the horse. My conjecture is that the term was from pugilism, predating horse racing. Boxing contests were often a series of contestants challenging the previous winner. The final winner would be the boxer that no one could beat. The first boxer could present himself for a match (take on all comers) but stand unchallenged due to his recognized superiority. He would have no need to raise his hands to fight and would, thus, win "hands down." I have not found an OED reference for "hands down" though "hands up!" is listed.