display taxonomy terms with link

You’ll need to use get_term_link(), e.g. change your $terms array to be HTML for the terms or links instead:

    foreach( $taxonomy_terms as $term ) {
        $term_link = get_term_link( $term->slug, $taxonomy );
        if (! is_wp_error( $term_link ) ) {
            $terms[] = '<a href="' . esc_url( $term_link ) . '">' . esc_html( $term->name )
                       . '</a>';
        } else {
            // Could not fetch link for this term
            $terms[] = esc_html( $term->name );
        }
    }