To perhaps explain the @enumaris's answer in simpler terms: when you curve spacetime, you lose the ability to define a universal "now", even one that is relative to a particular observer. That is, it is meaningless in general relativity - and thus, as far as we can tell to the best of our knowledge - in the real Universe - to ask "what is the happening 'right now' in the Andromeda Galaxy". The Andromeda Galaxy occupies its own time-stream, if you will, compared to ours, and this is also true (to a more limited extent) for the individual stars within a galaxy due to the differing curvatures of spacetime around them.
But this doesn't really answer the question of why the drive is able to "move" faster than light. The "faster than light motion" can be defined rigorously in a very simple sense - since the spacetime as described in the papers is exactly flat except for a certain region, it is possible to "globally" compare it against an identical copy (with all relevant "landmarks" included in both) with the distorted region flattened out, and you can show that there exists a path within the warp-drive space that, if followed, will take you from one locale of the flat spacetime to another in both a proper and a coordinate time smaller than that required for light to travel the distance in the corresponding flat spacetime.
Also, regarding causality concerns: in the ideal Alcubierre spacetime, they do not exist. However, when you add the option of having two Alcubierre bubbles (the distorted regions mentioned), or the ability to form and dissipate the bubble out of and into nothing more than once (i.e. a warp drive you can turn on and off, as one that you would want to use for realistic space travel would have as a desirable feature), then you can easily come up with solutions that will violate causality, and they effectively look the same as what you'd need to do in ordinary special relativity to send a message backwards in time using tachyons.
(Indeed, in the famous movie Star Trek IV, they would not actually have had to slingshot around the Sun to go back in time - simply-arranged accelerations together with warping would have sufficed to arrange for them to arrive at Earth before leaving.)
The relevance of the local vs. global distinction is specifically with regard to the question of "information transmitting at 120% of light speed" and this being a violation of SR. There is no conflict with the SR light-speed barrier because, as said, that is a purely local phenomenon: you will never observe yourself catching up to and striding past a photon on your immediate right (or left), no matter where you are in any spacetime or what you do. That is what is meant by a "local" violation of the light-speed limit.
"Global" quantities do exist in GR, just not global simultaneity, in even a relative form. (One famous example is the "mass of a black hole", which in the Schwarzschild concept is naively zero, because that spacetime is vacuum, that is, local energy density is always zero, even at $r = 0$! It's not a delta function, not a point mass - $T^{\mu \nu} = 0$ everywhere! The mass does not exist thus as a local property, but is strictly global.) You can still talk of the time elapsed for a trip to a distant locale, the round trip time of a light signal, etc. . And this also means that you can indeed "globally" violate causality with the Alcubierre drive (under the given stipulations of either multiple drives or forming/dissipating the warp field from and to nothing). While local properties arise as differentials, global properties arise as integrals (which is really the "essence of calculus" in some fashion: derivatives tell you everything you'd want to know about a function or space locally, meaning "in a vanishingly or ideally small region near a point", integrals synthesize the structure of that object over a wide area and let you obtain some statistic regarding the whole.).
That said, as mentioned in the comments these solutions are, sadly, almost surely unphysical. You are effectively arbitrarily stipulating the geometry, and when you do that the constraints that are needed for physical realizability can and will very easily pop: at the bare minimum for this one you need negative energy ($T^{00} < 0$) but if you are to do the form-and-dissipate of the bubble that is needed to make a practical drive, you worse need a priori the ability to create the very tachyonic signals you are thinking of! Negative energy is still a potentially fringe possibility - but unlikely(*) - however, the latter would essentially amount to having a good portion of the prize we seek (i.e. FTL travel)!
Of course, it may be possible that all this changes with a theory of quantum gravity; however it could also (and many would bet on it) change to be even more restrictive, not less.
(*) You may have heard "fringey" or "pseudosciency" claims that "zero point energy" could somehow be involved if you poked around on this enough. This is not correct, though this goes over to quantum mechanics, not general relativity. In quantum mechanics, there is the idea of a vacuum state of a quantum field such as those like the electromagnetic field which fill up the spacetime. This is the state with no particles in it, and is usually (but not always!) the lowest-energy state. A simple calculation assigns it tremendous positive energy, but a) you can easily reinterpret this as just a shift of reference level and take the energy zero with no harm at all to the theory's mathematics, and b) more to the point, creating a "negative" energy density would mean trying to find a field configuration with lower energy (thus under the reference point shift comes out as negative) to occupy the region of space in which you want the negative energy effect. Either this is impossible because no lower energy state exists, or catastrophically undesirable because to instantiate such a state would be tantamount to our universe turning out to be what is called a "false vacuum" and then collapsing it: endgame for everything within our future light cone, not a warp drive that would take us to regions outside it. In simple terms, the warp drive wannabe turns out to be in fact a doomsday bomb that kills us and keeps on killing with a wave of death moving out at the speed of light, and leaving uninhabitable space in its wake. (It kind of reminds me of another Star Trek episode called "Force of Nature" from Star Trek: The Next Generation, where a scenario in which a warp drive was imagined to do something like this over a suitably long period of use, or if catastrophically exploded (as they did) due to malfunction. The difference here is that this effect would happen with any warp drive, and happen at the instant it were switched on, making warp travel a complete no-go... sigh.)
