The 3-star rank is a rank in Mario Kart that is given to anyone who can complete every course with a 3 star rank. This means you'd have to do all 8 Cups in 50cc, 100cc, 150cc and Mirror Mode on 3-stars to get this rank. The advantages of having this rank? None, but it makes you look like you know all the tracks and Cups back-to-back. It also only shows up in Wifi and Multi-player mode, so its really only to "Show off" how "great" you are at Mario Kart.
There is no specific known method to achieving a 3-star rank, it will just take great driving skill, meaning you have to win every race by a good distance, and avoid track obstacles such as Goombas, or even hitting the edge/falling off.