The Sun as pictured above isn’t exactly lying, but what he’s saying is misleading. The nine eight planets of the Solar System do orbit the Sun, which may give you the impression that the Sun is stationary with respect to the planets. It is not. The Sun wobbles.
This is due to the combined gravitational forces of all the planets, comets, asteroids (and even Pluto) tugging the Sun in multiple directions at once. The effect is small but not insignificant. Over the course of just a few years, the Sun can completely change its position relative to the rest of the Solar System.
The true center of the Solar System is a point in space called the barycenter. The barycenter happens to be located near (and sometimes inside) the Sun. You could say the Sun, planets, and everything else in the Solar System orbit the barycenter, but keep in mind that the barycenter is not stationary either.
If you want to get a better feel for how planets and stars tug on each other through gravity, I suggest checking out the game Super Planet Crash. It’s free. Just click here to start playing.
One Way to Find a Planet from NASA’s “The Space Place.”
Barycentric Coordinates (Astronomy) from Wikipedia.
The Wobbling Sun from Innumerable Worlds.