The first set of tires you can buy i.e. that go to +3 are for driving on cement roads only and are not good for driving on courses with dirt roads. The second set of tires i.e. go to +2 and have a square in the middle are for dirt roads.
To activate this mode, drive in the town, go to the location of the drag race and hit the gas. Follow the markings in the middle of the road. You will arrive at a square with the "under construction" thing in the middle. Drive straight through that, it is a jump ramp. If you have a fast enough car, you will reach the clock tower at the top, and the mode will activated.