The UK Phrase Dictionary explains that the expression means “From the highest authority.”  Bettors are always eager to hear which horse is likely to win a race and are eager to hear what stable workers and trainers believe. “The notional ‘from the horse’s mouth’ is supposed to indicate one step better than even that inner circle, that is the horse itself.” The Syracuse Herald published a comment in May 1913, “I got a tip yesterday, and if it wasn’t straight from the horse’s mouth it was jolly well the next thing to it.”