For preventative maintenance, and to comply with the recommended replacement interval specified by the vehicle manufacturer. Rubber timing belts contain reinforcing cords of fiberglass or Kevlar. These cords keep the belt from stretching or breaking, which is no simple task considering the strain that is on the belt. Many other engines use a steel chain for this same purpose.
As the belt accumulates miles, the constant bending every time it passes around a pulley plus constant exposure to heat causes the rubber to harden and become brittle. Little cracks start to appear in the rubber. These cracks are not serious as long as they're not too deep or cause chunks of rubber to break loose. The constant stress, heat and bending also weakens the fibers that make up the reinforcing cords. Eventually the point is reached where something gives and the belt fails.
Until recently, the incidence of failure peaked sharply after about 60,000 miles, so the vehicle manufacturers mostly recommended replacing the best at 60,000 mile intervals. On newer engines, this interval has been extended to as much as 100,000 miles thanks to improved belt materials.
So what happens if you don't replace the belt? Maybe nothing. Some belts just keep going and going and will last the life of the engine. Others may fail at 60,001 miles or even sooner. Since no one can predict the exact mileage at which a timing belt will fail, the safest bet is to follow the vehicle manufacturer's recommendations.
WARNING: A timing belt failure can cause extensive engine damage on many overhead cam engines. If the engine lacks sufficient valve-to-piston clearance, a belt failure can allow the valves to hit the pistons destroying both. The cost of replacing the belt at the recommended interval, therefore, is cheap insurance compared to the cost of replacing or rebuilding the engine.
Engines with timing belts that are at risk for this type of damage include: