The article doesn't explain what the warp drive actually does. As far as I can tell the idea can be roughly phrased as "rather than making a long journey, cause the road in front of you to become short, then make a short journey". The drive would contract spacetime between the object and its destination to make it really small. Apparently to do this, one has to also affect spacetime behind the object, expanding a region of it. Once spacetime is distorted appropriately, the long journey becomes short. Seems vaguely plausible - as much as physics ever does.
On the other hand, the process of distorting space time should propagate at the speed of light at best. So the ship would spend a standard amount of time bending space-time and not moving, then move a short distance and arrive at the destination. And then after the trip, we'd still have all this distorted space-time to either fix or leave stretched.