The AF confirmation actually works on a very similar principle to the old focus screen, in that they compare two different optical paths in order to determine exactly how in or out of focus the image is. The exact technique is different, but they accomplish the same end goal. The only difference is a computer is comparing the sides of the circle for you.

In a PDAF (Phase-detect auto focus) system, the camera is looking for the light entering the sensor to have the right relationship to each other. When it isn't (such as when the circles didn't line up), it knows that things are out of focus. Once the phases line up properly, it can confirm that the image is properly focused on whatever is under that focus point.

It normally communicates this by blinking or lighting up the focus point or by beeping. The focus confirmation chip provides a few key details about the lens that allow for this process to be completed.