The Brass Ring was originally a game reward or prize related to riding a carousel. Most carousels did not have "jumpers" - animals which move up and down as the carousel spins - The outer row animals were still/fixed. Many riders of carousels tended to ignore the outer rows in favor of the jumping animals. In order to encourage more riders to sit on the outer rows, someone devised the ring game and it became a common fixture of the carousel.
The object of the game was simple. A rider sits on the outer row and when the carousel spins around, the rider reaches out and grabs one of the metal rings from the dispenser. Since not all the rings were brass - typically one brass ring per ride among many non-brass rings - getting the brass ring was a special treat. The brass ring usually entitled the holder to a free ride when they returned the brass ring to the attendant. Or could be kept as a good luck charm, memento, etc.
It's important to note that in their heyday, most carousels were pay-to-ride amusements. Back when a couple pennies, a nickel or a dime was still significant change, a free ride on the carousel would seem much more relevant. Therefore, to get the elusive brass ring was to win a notable prize equated with fun, gratification, fulfillment, etc.
Little Amy went to the park today and rode on the carousel. On the first try, she got the brass ring. Amy now could ride the carousel again for free. Amy was very happy.
Amy grew up to be a pleasant, pretty, hard-working young woman. She got married, started a family, and lived in a nice modest home with her husband. Together they worked towards prosperity and raised a good, healthy family. Amy was very happy. Again, Amy got the brass ring.