Why is the exhaust mounted to the vehicle by flexible joint?

Thermal expansion.

An exhaust pipe gets up to 600 degrees or more. The coefficient of expansion of steel is 8 millionths of an inch per inch per degree. a 12 foot exhaust increasing (and decreasing) in temperature by 500 degrees will expand overall a little more than 1/2 inch. If the pipe were mounted solidly the mounting bracket would eventually break, or the pipe would break, from the strain put on it by the expansion.