After a number of events you complete during your run to upgrade your license, the game shows you a car and says that the car is now around Paradise City. Let's call those Burning Route cars.
What you need to do is take those cars down, similar to doing it on a Road Rage.
The way to make those kinds of cars spawn is simply by driving around. Eventually you'll see the car in the distance because it moves faster than normal traffic, and more irratic. If you successfully take it down, it becomes available for you in any junk yard (as the game tells you).
From that point, head to a junk yard and choose that car. Now, press the Back button, which takes you to the in-game map. From there, make sure you're looking at "Events". Select the type of events, which defaults at "All", and move it left or right, until you get the selection for "Burning Route".
From there, as long as you have a Burning Route car, your map should show one out of all the burning routes as a white star. Head to that one, which should also be labelled as a burning route for your specific car.
Simply win a burning route for each Burning Route car to unlock this achievement. A burning route is a single race from one point in Paradise City to another.
If you highlight the burning route in the map, it will tell you the starting and end points of the burning route before you go in. Once you learn the city, you'll be able to plan your course before even getting to the burning route.
Now, during your game, you'll want to do these burning routes as soon as you get their cars, because they count towards your license upgrade. By the time you upgrade to Burnout license, you'll have 8 Burning Route cars left to get.
After winning a burning route, you get an announcement that informs you that you have received a special version of that same Burning Route car in your junk yard. These special versions are simply for your own use, so enjoy them.
The only way to play burning routes is to use the cars that naturally get announced as entering the city and take them to their proper burning route event.
*EDIT* There should be 35 burning routes in Paradise City, and 1 burning route in the DLC island.
Good luck! You'll be doing this for most of the game.
If you'd like more information, leave a comment and I'll gladly help with info.