The features Live streaming and Embed live streams need to be enabled in https://www.youtube.com/features . Please follow the instructions there.
To embed into your website, a linked AdSense account seems to be required.
Also refer to:
Set the stream to unlisted or public
Go to the “Watch page” (View on watch page link (lower right) in https://www.youtube.com/live_dashboard)
Click Share -> Embed to get the embed HTML code
Now the problem is: every time the stream stops and restarts, the ID changes.
To get a permanent URL, you need to change the link to use the Channel ID (12345 in this example):
Get the Channel ID from https://www.youtube.com/account_advanced. Not the User ID!
Add this into your website HTML, and open the website in anonymous browser window to check if the stream displays without error message.