Why two minutes
The American Dental Association's home oral care page reports that brushing for two minutes was associated with a greater reduction in plaque than brushing for one minute, and that brushing twice a day was best at lowering cavity risk. A timer makes the full two minutes easy to hit, because most people, and nearly all small children, stop sooner than they think.
Counting 120 seconds is hard for a child, but 30 seconds is a short job.
Counting 120 seconds is hard for a child, but 30 seconds is a short job. The timer walks the mouth in a fixed order: upper left, upper right, lower left, lower right. Each quarter has its own colour (teal, blue, lilac, pink) and its own two-note sound, so a child who is not looking at the screen can still hear that it is time to move.
Reading the mouth picture
The ring in the middle is the mouth as you would see it in a mirror: the top half is the upper teeth, the bottom half is the lower teeth, and left is the left of the picture. The quarter to brush now has a bold outline and a solid colour. Finished quarters stay softly coloured and the ones still to come are pale.
Where the timing comes from
Source: American Dental Association, Home oral care, https://www.ada.org/resources/ada-library/oral-health-topics/home-oral-care (two-minute brushing and twice-daily brushing findings). The split into four 30-second quarters is a pacing choice made for this page, not an ADA requirement. This page is a timer, not dental advice. Ask your dentist how and how often your child should brush.