The reasons for female genital mutilation differ:
It can be done out of religious conviction (mainly by Muslims).
It can be a rite of passage and proof of adulthood: she is no longer a girl but a woman.
It can be an expression of superstition. For instance the belief that the clitoris is poisonous, that it can cause a child to die if it comes into contact with the clitoris during delivery or can cause a guy to die or become impotent by contacting it.
It can be done for aesthetic reasons (a flat vulva or the belief it will give the woman a more beautiful face).
It is often done to ensure virginity and chastity. By removing (part of) the external genitals they severely damage or destroy the ability to enjoy sex.