If you're wondering what Kart Racing is, it is exactly as it sounds. You drive Karts, a type of mini car designed for a mini track.
To unlock this, you'd need to complete either the hero side mission Route 101 (Tail's Stage) or the dark side mission Route 280 (Rouge's stage). Completing both sides results in unlocking multi-player racing, where you race each other to the finish line. In single player racing, however, you race AI. But before racing the AI, you have to choose your character before you race.
There are six characters, which are the ones you play in the story mode: Sonic, Tails, Knuckles, Shadow, Eggman, and Rouge. Choosing any of these six will take you to the tracks. There are three tracks: Beginner, Intermediate, and Expert. Depending on the track's difficulty depends what it will look like, and how tough it will be (Obviously). Completing any of these tracks awards you an emblem.