To get this you must be playing in a team race (eg. Sonic, Tails and Knuckles all in one team whilst you control one of the characters). When trailing a teammate you will see a golden line behind them, prompting you to slipstream them. Simply drive on the line and charge up the "Slingshot" then upon exiting the stream you will receive your speed boost.
Upon completing the action correctly, you will notice some text to the right of your vehicle with a golden star writing "Slingshot". This means it has registered your Slingshot!
Do this 10 times in a SINGLE race and the achievement will be yours.
I created this video to help:
Hope this helps.