Why do we say "straight from the horse's mouth"?

It's another racing term. When looking for information on a horse's form, you might want to get to a source close to the horse itself, such as the trainer or, sometimes even better, the groom that looks after the horse.

Thus, a tip that came straight from the horse's mouth implies that it came from even closer to the horse - the horse itself. It originally meant a very good tip to work out a bet, but then extended to become something you might say about any very reliable piece of information.