IN that link, it states: If you already have a WordPress.com account, your API key is listed on your profile page, which you can get to by clicking the “My Account” link in the top right when you’re logged in.
But when I go to users, Your Profile, I dont see the API?
Regardless, the youtube api key only applies to WP.COM accounts, not self hosted account unless it is for Akismet or WPstats, not for embedding youtube (no idea why). You will need the above plugin as I mentioned.
Also, to get the API key you not only have to register at WP.com you have to start a blog there too. Well, you will own the whateveryouwant.wordpress.com name, but don’t have to post anything there.