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 that moving not straightly forward is highly descouraged because of the damage done to the ships hull.

You may picture it as if there would be a strong gravitational force in front of the spaceship which draggs it through the room.

![Image of the NX-01s warpfield](http://images.wikia.com/memoryalpha/de/images/9/97/USS_Enterprise_NX-01_und_USS_Columbia_NX-02.jpg)
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).

(Please help me correct the not-true-english-techtalk errors, I watched most episodes in german and only recently started to rewatch them in english, thx!)

**update**:
added a picture of the nx-01 and its warp field with explanation