Add shipping method description from Shipping Zones backend, ant print it under chosen shipping method in front.

woo_shipping_zone_description.php

//
/**
 * Check if WooCommerce is active
 **/
function smdfw_is_woocommerce_activated() {
    return class_exists( 'woocommerce' );
}

/**
 * Init plugin
 **/
add_action( 'init', 'smdfw_init', 100 );
function smdfw_init() {
    if ( ! smdfw_is_woocommerce_activated() ) {
        return;
    }
    // Add shipping methods filters
    $shipping_methods = WC()->shipping->get_shipping_methods();
    foreach ( $shipping_methods as $id => $shipping_method ) {
        add_filter( "woocommerce_shipping_instance_form_fields_$id", 'smdfw_add_form_fields' );
    }
}

/**
 * Display admin notice if WooCommerce is not active
 */
add_action( 'admin_notices', 'smdfw_requirement_notice' );
function smdfw_requirement_notice() {
    if ( ! smdfw_is_woocommerce_activated() ) {
        /* translators: %1$s: open link, %2$s: close link */
        $error   = sprintf( __( 'WooCommerce Shipping Method Description requires %1$sWooCommerce%2$s to be installed and active.', 'smdfw' ), '<a href="http://wordpress.org/extend/plugins/woocommerce/">', '</a>' );
        $message = '<div class="error"><p>' . $error . '</p></div>';
        echo $message;
    }
}

/**
 * Add description field to shipping method form
 */
function smdfw_add_form_fields( $fields ) {
    // Create description field
    $new_fields = array(
        'description' => array(
            'title' => __( 'Description', 'smdfw' ),
            'type'  => 'textarea',
        ),
    );
    // Insert it after title field
    $keys  = array_keys( $fields );
    $index = array_search( 'title', $keys, true );
    $pos   = false === $index ? count( $fields ) : $index + 1;
    return array_merge( array_slice( $fields, 0, $pos ), $new_fields, array_slice( $fields, $pos ) );
}

/**
 * Load description as metadata
 */
add_filter( 'woocommerce_shipping_method_add_rate_args', 'smdfw_add_rate_description_arg', 10, 2 );
function smdfw_add_rate_description_arg( $args, $method ) {
    $args['meta_data']['description'] = htmlentities( $method->get_option( 'description' ) );
    return $args;
}

/**
 * Display description field after method label
 */
add_action( 'woocommerce_after_shipping_rate', 'smdfw_output_shipping_rate_description', 10 );
function smdfw_output_shipping_rate_description( $method ) {
    $meta_data = $method->get_meta_data();
    if ( array_key_exists( 'description', $meta_data ) ) {
        $description = apply_filters( 'smdfw_description_output', html_entity_decode( $meta_data['description'] ), $method );
        $html        = '<div class="shipping_method_description"><small class="smdfw">' .  wp_kses( $description, wp_kses_allowed_html( 'post' ) ) . '</small></div>';
        echo apply_filters( 'smdfw_description_output_html', $html, $description, $method );
    }
}