The Grease Monkey booster increases all of your car's stats for a single race. For instance, if you have a 100HP car, it will make your car behave like it's a 250HP car. The boost only lasts for that race.
To use the Grease Monkey, click the button that appears on the left of the screen at the start of the race. If you have any Grease Monkeys available, it will be applied instantly. If you have run out of Grease Monkeys, you'll get a chance to buy more.