You’ll want to stick something like this in your theme’s category.php
file:
<?php
// fetch the categories with the parent being the current cat
$sub_cats = get_terms('category', array(
'parent' => get_queried_object_id()
));
if(!is_wp_error($sub_cats) && $sub_cats)
{
foreach($sub_cats as $cat)
{
// Create your grid here, this is just an example
printf(
'<a href="https://wordpress.stackexchange.com/questions/57529/%s">%s</a>',
get_term_link($cat, $cat->taxonomy),
$cat->name
);
}
}