How can I display a list of categories of my posts

You need to use wp_list_categories() (Codex ref), using the child_of argument.

Assuming you know the ID of the “Podcasts” category, e.g. 123:

wp_list_categories( array(
    'child_of' => '123'
) );

If you need to find the category ID, use get_category_by_slug (Codex ref):

$cat = get_category_by_slug( 'podcasts' );
$catid = $cat->ID;

wp_list_categories( array(
    'child_of' => $catid
) );

Rather than executing PHP from within the Page content, I would simply create a custom Page template:

1. Copy page.php as page-podcast-list.php
2. At the top of page-podcast-list.php put the following:

.

<?php
/*
Template Name: Podcast List
*/
?>

1. Look for the post content markup (e.g. <div id="post-<?php the_ID(); ?>" <php post_class(); ?>>), and replace what’s inside it (should include <?php the_content(); ?>, etc.) with the $wp_list_categories() code from above.

If you have problems, copy/paste the code from page-podcast-list.php in your answer, and we’ll help from there.

EDIT 2

Your parse error is here, inside of your wp_list_categories() argument array:

'child_of' => $catid;

Should instead be:

'child_of' => $catid

(no semicolon)