You get so many respondents, you can give them all different combinations. Each race you disregard the losers, and when the population gets low enough you might miss a race, you just say "hey, I predicted x races, call Bob to confirm".
Yes, that's basically how it worked. To be clear, Brown was careful to promote and run it for entertainment and not a scam. He made very clear up front that those choosing to participate were very likely to lose their money and insisted that the early bet amounts were minimal "fun money". The ever-shrinking number of those continuing to later rounds had made enough from earlier wins to offset the later eventual loss.