We are in the fairground, specifically on a carousel or
merry-go-round. At one time, the riders on the outside row of horses were often given a little challenge.
Once the ride started moving, a metal arm was swung out — on some rides this held a single brass ring, which riders could try to grab as they passed. Anyone who managed to retrieve it could redeem it for a free ride. Another system had a dispenser of rings, most of which were steel and had no value, but one per ride was the brass one that won the prize.
“Brass Ring” came to have the figurative sense of a prize, in particular one that was hard to gain. Grabbing the brass ring, going for the brass ring or reaching for the brass ring were all used to refer to the opportunity to compete for a grand prize or to achieve ones goals.