In order to use this feature, you will need to make sure that you have "Embed Live Streams" enabled in your YouTube account. If you have a Google account, then this will log you in to YouTube as well. If you do not have an account then you can sign up for free.
To enable Live Streams, log in to YouTube and go to "Creator Studio" from the user menu in the upper right corner of your browser window:
Then, click on "Channel" (on the left side), and select "Status and features":
If your account is verified, you can enable the option to Embed Live Streams:
If your account is NOT verified, you'll need to verify your account before you can enable Live Streams. To do this, go to https://www.youtube.com/verify, and proceed from there.
Now that Live Streams are enabled for the YouTube account you want to use for broadcasting, you can create a Live Event with Reflector Teacher. When you click the "Broadcast" button for the first time, you will be prompted to sign into your YouTube account via a popup window. More details on this process can be found here: How do I use Live Streaming in Reflector Teacher?
Once you have completed this login, you will be able to start broadcasting. To see your Live Streams in your YouTube account, go back to the Creator Studio, and click on "Live Streaming" (on the left side of the screen), and select "Events":