Freud was trained as a medical doctor at the University of Vienna, and went on to specialize in the treatment of nervous diseases. Yet, Freud was such a good writer and storyteller that he won the Goethe Prize for Literature—a prestigious literary honor won also by novelist Thomas Mann and film- maker Ingmar Bergman. Freud gave his case stories imaginative characters with names like the "wolf man," "the rat man," and "little Hans" (a boy who had a horse phobia). Freud's most famous story was of course a famous re-make of the tale of Oedipus, which he adapted from the Greek playwright Sophocles.
Freud knew how to capture his readers' imaginations as well as their intellects which is the essence of being a premier storyteller. Besides case stories, he also wrote volumes of theoretical papers but, at least for me, his stories are more memorable and more useful than his theoretical constructs.
I believe that Freud presented his clinical cases in the form of stories because stories have a closer connection to nature and human nature than the categories of logical reasoning. Mind, especially unconscious mind, lends itself to stories. Stories come closer to capturing the essence of human experience than the categories of classification in, for example, the various editions of the Diagnostic and Statistical Manual of Mental Disorders. DSM Diagnostic categories are pegged to individuals; whereas stories are about the interactions between people--the real locus of human problems.