WordPress get child terms of current taxonomy page and display them.

get_child_terms.php

<?php

$taxonomy_name = get_queried_object()->name; // Get the name of the taxonomy
$term_id = get_queried_object_id(); // Get the id of the taxonomy
$termchildren = get_term_children( $term_id, $taxonomy_name ); // Get the children of said taxonomy

// Display the children 
echo '<tr>';
foreach ( $termchildren as $child ) {
  $term = get_term_by( 'id', $child, $taxonomy_name );
  echo '<td><a href="' . get_term_link( $term->name, $taxonomy_name ) . '">' . $term->name . '</a></td>';
}
echo '</tr>';

Generally, this works for me pretty well.

<?php
$parent = get_queried_object();
$children = get_term_children($parent->term_id, $parent->taxonomy);
if (!empty($children)) {
    echo '<ul>';
    foreach ($children as $child) {
        $child_term = get_term_by('id', $child, $parent->taxonomy);
        echo '<li><a href="'.get_term_link($child_term, $parent->taxonomy).'">'.$child_term->name.'</a></li>';
    }
    echo '</ul>';
}
?>

However, if sorting child terms is important to you then you can go through a different route as well.

<?php
$parent = get_queried_object();
$children = get_terms(array(
    'taxonomy' => $parent->taxonomy,
    'parent' => $parent->term_id,
    'orderby' => 'slug',
    'order' => 'ASC',
    'hide_empty' => true));
if (!empty($children)) {
    echo '<ul>';
    foreach ($children as $child) {
        echo '<li><a href="'.get_term_link($child).'">'.$child->name.'</a></li>';
    }
    echo '</ul>';
}
?>

Hope this was helpful to some. Thanks.