How to show Sub Categories on Categories page?

You can use get_categories & wp_list_categories with specific parameters (to use directly in you category.php template):

// List sub cats
$params = array(
    'parent'        =>  get_queried_object_id(), //id of current category displayed 
    'orderby'       => 'name',
    'order'         => 'ASC',
    'hide_empty'    =>  false // do not hide sub cats without posts
);
if ( count( get_categories( $params ) ) ) {
    wp_list_categories( $params );
}

Full list of arguments can be found here

You can also customize the output by user your own Walker extending WP’s Walker_Category class.