Collecting Gold Rings fills your Ring Boost Bar which increases your runner's top speed, up to a maximum of 100 Gold Rings. The more Gold Rings you have the faster your top speed will be, but be careful: taking damage will cause you to drop the Gold Rings you have collected. During a race, you can see your current Gold Ring total at the top-right of the screen.
After the race ends, you keep the rings you've collected and can use them to upgrade your runners.