At any point during the main story mode of the game (not Time Attack).
Every time you crash or bump something, your car takes damage, which is tracked at the top right of the screen during races. You get the opportunity between races (and If you total your car mid-race) to repair this damage.
You need to spend 10,000 coins on damage repairs in total. Despite the achievement description, this does not have to be done in one single repair.