I agree that it is a confusing analogy. This is my understanding of it:
Normally, in a horse race, there are many rules that the jockeys must follow. The adjudicators may hold a (quick!) investigation, called a Stewards' Enquiry into any claims of that rules have been broken, and decide to disqualify a horse, or to adjust the rankings based on some penalty for failing to follow the rules.
In First Past The Post rules, however, there is an agreement between the bettors that they will disregard the stewards' tweaks, and instead base the decision "purely" on which horse passes the winning post first.
Many voting systems, such as Single Transferrable Vote, involve finding a provisional ranking of several candidates, eliminating a losing candidate, and transferring the losing votes to the next preference, which changes the rankings. The candidate with the most primary votes may not be the final winner.
By analogy, this is like the horses being re-ranked by the stewards. [Not for infringing any rules, I should add.]
However, First Past The Post voting systems have no such ranking adjustments. It is the least complicated system - the most primary votes - just like least complicated system in horse racing - the first horse to reach the finish line.