I've seen this movie:

It shows Sun-synchronized orbit of a satellite. What causes this orbit to be Sun-synchronized? I mean, from what I know, it shouldn't rotate as it does. Twice a year it should be perpendicular to the Sun, and twice a year it should be parallel. Apparently I'm wrong, so why does it rotate?

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Sun-synchronous orbits are those whose orbital plane makes a constant angle with the radial from the Sun. For that to occur, the orbital plane of geocentric satellites must rotate in inertial space with the angular velocity of the Earth in its orbit around the Sun, which is $360^\circ$ per $365.26$ days, or $0.9856$ degrees per day. With the orbital plane precessing eastward at this rate, the ascending node will lie at a fixed local time, or equivalently, at a fixed direction relative to the Sun-Earth radial line.

As pointed out by @Erik, this precession is due to the oblateness of the Earth (the Earth's equatorial bulge). The rate at which the line of nodes moves owing to this bulge is given by $$ \dot\Omega = -\frac{3}{2}J_2\left(\frac{r_E}{ℓ}\right)^2 n\cos\iota $$
where $J_2$ is the is the zonal harmonic coefficient ($1.08262668\times10^{-3}$ for Earth), $r_E$ is the body's equatorial radius ($6,378,137$ m for Earth), $ℓ$ is the orbit parameter (the semi-latus rectum), $n$ is the mean motion, and $\iota$ is the inclination of the orbit.

For a given orbit parameter ($ℓ$) and mean motion ($n$), the inclination of a geocentric satellite orbit can be selected to obtain a Sun-synchronous orbit.

Sun-synchronous orbits are possible because the Earth is not a perfect sphere. The Earth is best described as an oblate spheroid. This non-spherical shape and the resulting gravity field can be used to alter the orbit in a predictable way ― specifically to precess the orbit with a period of one year.

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