How can I display the category descriptions below the category name using a functions.php filter?

A lot of themes have this built-in so first try adding a category descriptions and see if that solves it. The next way to look at handling this is by modifying the taxonomy archive template by simply adding this below the title in the template:
‘, ” );
?>

But your question says specifically to add it by modifying functions.php. I haven’t tested this but this looks like it work (from: https://generatepress.com/forums/topic/adding-category-description-to-custom-post-type-archive/):

add_action( ‘generate_after_archive_title’, function() {
if ( is_post_type_archive( ‘video’ ) ) {
the_archive_description( ”, ” );
}
} );

It’s set to only apply to a custom post type of ‘video’ but you should be able to modify it.