WooCommerce archives (shop, categories, and tag pages) don’t display short descriptions by default. If you want this feature, customize your theme by adding this line of code:

add_action( 'woocommerce_after_shop_loop_item_title', 'woocommerce_template_single_excerpt', 5);

Place this code in the functions.php file of your currently active WordPress theme (or child theme), and you’re good to go.