Mopeds require gas to run. On the right hand side of the game screen, you will notice a fuel bar when you're riding your moped. If that is running low, be sure to refill your gas tank.
To fill up your tank of gas, ride with your moped to the station that is located right behind Verooom!! The platform will have the words 'ReFill Gas' on it. Ride your moped onto that platform and wait until a popup shows up, asking if you will like to refuel your moped.