To earn this achievement that's called "Racing Transformed", you must go All-Star.
All-Star is a random item that you earn from one of the question mark boxes you come across during any idea. If you hear the announcer go "All-Star", that means that you have earned the item. To use it, press A, then A again and you will go "All-Star". I did this with Sonic after going through a couple of races, the third race finally netting me the All-Star item and using it during the race. The moment you go "All-Star", that's when the achievement should be unlocked.
Have fun and good luck!