Why do the spoilers move in this video?
Two reasons that are really one reason. In short: Because lift.

In this case though, it's "to get rid of lift". The spoiler's job is to reduce the amount of lift the wing experiences. This can be done symmetrically, on both wings at the same time, to increase the rate of descent or keep an aircraft from lifting back off the ground when in ground effect, or in this case, it can be done asymmetrically, or only on one wing.

In order to turn, an aircraft has to create an imbalance of force, more on the side outside the turn, less on the inside of the turn. The wing to the inside of the turn needs to drop, but it doesn't really want to...it has lots of air pushing it up (when it doesn't, you will have a very bad day)!

The spoiler in this case is just acting as an extension of the aileron. By moving upward with the aileron itself, it spoils the airflow over the top of the wing a bit, reducing the lift, and causing the wing to drop a bit, raising the other wing. It also acts as a larger surface area for the incident airflow to impart force on, which also pushes the wing down.

On the other side, the aileron is doing the opposite, increasing the wing's effective camber, at least near the tip, and increasing lift. Flaps do the same, but also are usually used symmetrically to increase lift on both wings. In some airplanes, you can have a 'flaperon', which can also be used as an additional control surface while also doing it's flap-thing.