I couldn't figure this one out. Has anyone? I know it's about anti-matter and some kind of reactor, but what is it supposed to be doing?
The matter-antimatter reaction is just the energy source for the work it has to do. The energy created by the reactor is used by the warp gondolas to build the warp field: this deforms the space-time-continuum.
So the spaceship itself isn't accelerating; the warp field makes the ship "falling" into the direction it should go. This has some side-effects, as moving in a direction other than straight forward is highly discouraged because of the damage done to the ship's hull.
You may picture it as if there would be a strong gravitational force in front of the spaceship which drags it through the room.
As you can see, the time-space-continuum in the front of the ship is compressed, where in the back it is stretched. In the light gray area the normal room isn't altered. As a side-effect a observer outside of the warp field sees the ship stretched (like at the end of the intros).
This type of warp drive is known as an Alcubierre drive, proposed by the Mexican physicist Miguel Alcubierre. It is a speculative method of space travel that involves generating a warp bubble/ring that contracts space in front of the craft and then expands it again behind it. The Alcubierre drives moves space around the spacecraft, which remains stationary inside the bubble, resulting in the craft arriving at its destination faster than light would in normal space.