The icon on your wife's phone is probably a widget, sort of a mini-app on a home screen that does one simple thing or links to a larger app. I'm not sure the S4 actually has this specific widget available.
But what YouTube is saying is this: Start from the Home screen. In the lower right corner is the Apps button. Tap that, then on the next screen tap Widgets at the top. This will give a long (maybe very long) list of available widgets. Scroll that list to the right until you see an icon for Settings Shortcut, a blue gear. Tap and hold that. You'll see a kind of outline of your Home screen, and when you do, release the hold. Then comes a list of available settings that can be added as widgets on your Home screen. The list on my phone is fairly limited, though, and I don't see one for Auto answer, or for phone functions in general. However, your list might be different. If you see what you want, tap that, and the corresponding widget will be added to your Home screen.