To unlock the Speed Up! trophy in Cities: Skylines, you will first need to purchase and install the Snowfall DLC.
One of the new buildings available with the Snowfall DLC is the Road Maintenance service, which will be available once you reach the Busy Town milestone at a population of 1,200 - 5,000 depending on the map you choose to start on.
Position the Road Maintenance building next to a road and after a while, the trophy will unlock. You can unlock the trophy quicker if you add more Road Maintenance services and obviously if you fast forward the time to the max.