The cars will start moving after you complete the Superzilla quest and build the Bus Station.

Also, make sure that the animation option is turned on in the game's settings. It will be Yellow when it is turned on.
If you have the graphic quality set to low then you will need to raise the quality in order to turn on animations for the game.

See picture below:

Also please note: in order for the cars to drive around the city, there should be roads in your city. They can be purchased in the game shop, in the "Roads and Decorations" section - the "Fences and Roads" tab.