Exposed filters and contextual taxonomy description
You may use tokens (replacement patterns). If you add the term description in your fields and exclude it from display, you will have access to [description] token in a Global: Text area, for example, added in the view header. Then, if you check Use replacement tokens from the first row and set your exposed form style on Input required, when a category is selected, results will be filtered and the term description related to this category will be displayed in the header.

EDIT : for Drupal 8 (sorry, still more a "7 guy" ^^), the token follows Twig syntax : {{description_value}}