This should work, even outside of the loop:
$latestPost = get_posts('numberposts=1&category=YOUR_CATEGORY_ID');
The WordPress function get_posts sorts by post date, newest-to-oldest by default, so you’d get the newest post in whatever category you choose. Also, even though you are only requesting one post, the response will still be an array:
echo $latestPost[0]->post_content;
Edit:
Since the video embed code is being provided by a plugin that uses a shortcode, you’ll have to use the function do_shortcode on the content. If there is text (content other than the video) you’ll have to parse out the shortcode from the post_content.