Outputting subcategory title AND items on Category page

https://developer.wordpress.org/files/2014/10/template-hierarchy.png

Here you can view template file name that’s best fits for you.

https://codex.wordpress.org/Function_Reference/get_term_children

Use these function to get current term ( category ) childs. Note that in taxonomy.php template you can use $wp_query->get_queried_object_id(); to get current term_id you’r viewing

Use new WP_Query to echo posts inside the child terms loop to get posts from child term new WP_Query( array( '{taxonomy}' => {child_term_id} ) );

Here’s an example:
taxonomy.php ( or better category.php if it’s category taxonomy your using )[missing get_header() and get_footer()]

global $wp_query;
$cat_id = $wp_query->get_queried_object_id();
$cat_childs = get_term_children( $cat_id, 'category' );
if( $cat_childs ) {
  foreach ( $cat_childs as $cat_child ) {
    $child = get_term_by( 'id', $cat_child, 'category' );
    $child_posts = new WP_Query( array(
      'post_type' => 'post',
      'posts_per_page' => -1,
      'tax_query' => array(
        array(
            'taxonomy' => 'category',
            'terms'    => $child->term_id,
            'include_children' => false
        )
      )
    ) );
    if( $child_posts->have_posts() ) {
      echo '<h1>' . $child->name . '</h1>';
      while( $child_posts->have_posts() ) : $child_posts->the_post();
        the_title();
        echo "\r\n";
      endwhile;
      wp_reset_postdata();
    } 
  }
} else {
 // there's no category childs
}