From Sonic Retro
In Sonic Colours, collecting them unlocks more stages in the extra level Game Land, which helps to get the Chaos Emeralds and unlock Super Sonic. In Generations, collecting Red Rings unlocks new artwork and music for the Collection Room. This music can also be played in other stages over the normal stage music. In Sonic 4, only one red ring exists in each act, and collecting them all adds an achievement.
Colours was the first Sonic game to feature collectable red rings. However, the same concept was employed in Sonic and the Secret Rings, where Fire Souls (usually three per stage) could be grabbed to unlock extra Storybook pages or multiplayer modes.