Ambassador Showdown - Umamusume: Pretty Derby (Ignideus) vs. Heroic Sword (DifferentFight) 

Rush , racing , evolve you can Attack engage follower ... With storm you can do both Attack follower or opponent Leader

I'll explain it since you probably didn't understand their comment well. Essentially like you said, racing is similar to evolving where you spend a carrot (By placing a facedown carrot in the Evolve Deck underneath the Uma follower you are going to race) and 1PP in the Evolve Deck to race an Uma follower (Since racing applies to evolving, you can only Race once per turn or Evolve once per turn). Racing gives the Uma follower Rush and it can also proc an additional effect that some Uma followers have when they start racing. For example, when Fine Motion starts racing, it gets +1/+1 and can gift your leader +2 defense as well as milling the top 2 cards of your deck.