The colloquial explanation is that the spacetime in front of a ship contracts and the spacetime behind expands. I see how one could think that this would bring you closer, but I don't see that it actually does.
Let's look at the front of the bubble. So spacetime "contracts". If you look at a differential volume being run over, I would think it contracts to a minimum, but then must expand again to pass under the ship. If the bubble is moving, but the ship inside is not accelerating (let's assume it's also motionless in its space), then spacetime must be "piling up" in a (compressed) region in front, because you can't throw away the extra spacetime.
To paraphrase A Wrinkle in Time's analogy: think of some fabric and a bug that wants to get from where it is to somewhere else. You can fold the fabric and jump across the gap, but what I'm thinking of the Alcubierre drive doing is "scrunching up" the fabric: the bug still needs to walk over the same length of material, it's just that it looks like less distance.
Is all the above bogus? I don't really know what I'm doing with the physics here, as I have more of a mathematical background. I'm not seeing a way around this without introducing some discontinuity in space, which (I don't think) was the point. I also read somewhere that the expansion/contraction analogy could be misleading; is this why?