The mutation operator is very important for both continuous and enumerated chromosomes because it injects new genetic information into the population.  It is worth mentioning that the whole process of evolution depends to a great extent upon the initial genetics of the population.  The crossover operator does most of the work:  it selects the best mates which in turn pass their best qualities to their children.  Unfortunately, crossover does not contribute to the emergence of new qualities; it simply develops existing ones.

Without mutation the population has a high probability of degenerating.  This means that all individuals in the population will have identical genetics.  It is fine if this state is optimal, but in most cases the degeneration does not lead to optimal genetics in terms of the fitness function.

An example from medieval history of some British and French royal families can illustrate this point.  Closed evolution (inbreeding), which occurred during centuries when marriages were only allowed between relatives, led to the degeneration of their offspring in many cases.

An offspring is not a direct copy of its predecessors.  In order to meet this requirement in continuous chromosomes, some 0s are randomly changed to 1s and vice versa.  This process of randomly modifying chromosomes is called mutation.