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Damage in racing games generally falls into two varieties - cosmetic and performance-affecting. Cosmetic damage came first due to hardware limitations and often the damage was minor in nature - cracked windows and slight dents even after driving your car straight into a barrier at 140mph.
As racing games grew more sophisticated they began modeling damage on the essential components of a car - the engine, transmission and steering - depending on where and how hard you hit your vehicle. Impacting on the front might affect steering and have the car pull to the right, and getting slammed from behind could affect the transmission, causing sluggish shifting over the course of the race. Most simulation-focused racers allow you to pit during the race, which (usually) fixes that sort of crippling damage. Most of them also allow the player to switch off either form of damage depending on their preference or skill level.
Damage can be visualized more severely with generic cars, however. Most automotive manufacturers get skittish when certain aspects of damage come up - except for games with a strong focus (like rally racers) you'll never see licensed cars roll over, break in two, or crumple so severely they're incapable of driving. The crashes seen in the Burnout series would be impossible to pull off with licensed cars as the companies would never allow it.