How do you embed a video into a custom theme on WordPress?

I’m using WordPress 4.2.3 and I created a custom theme to use.
When I try to embed a YouTube video (pasting the http:// link into the content area), the video shows up in the edit content area, however when I view my site, it shows up as a text url instead.

I tried all sorts of things like clear formatting, removing link, unchecking settings in the Writing area, etc.
I temporarily switched to a default WP theme and the video shows up, so I’m definitely missing something in my template code, and I can’t figure out what.

Here is the code I used for outputting content in my template.

Index.php

<div class="video">
 <?php
  $post_id = 26;
  $queried_post = get_post($post_id);
  echo $queried_post->post_content;
 ?>
</div>

Thanks so much for the help – I really appreciate it!