His screenshot was taken from the user admin page under the Groups section.
If you go to a user’s public profile as an administrator, you should see the 
button near the top right. Scroll down a bit after you have selected that button. You should see the “Groups” section and the options in that screenshot.
You can also set up a group so that it will automatically be set as the primary group of all its members. If you go this route, just be aware this automatic feature might change any existing primary group settings of its members.