The Alcubierre drive does not break special or general relativity. It arises from some speculation on forming and maintaining a solution of the General Relativity equations found by Alcubierre, that presumes the existence of negative mass/energy density.
See the wiki paper on it at https://en.m.wikipedia.org/wiki/Alcubierre_drive
General Relativity allows you to solve for the metric, and geometry of spacetime, given the right hand side of the equation, which is the stress energy tensor (and either some initial conditions or other conditions or symmetries). But the stress energy tensor, i.e., the material source of the mass/energy (and momentum and stress) is not specified by General Relativity. It comes from whatever the type of matter/energy it is. A simple way to model it is using a model of material with energy and mass density, and so on, such as in hydrodynamics, kinetic theory or fluids or gases. You can also put in a classical or quantum field stress energy tensor on the right. Postulating a fluid with negative mass energy is just that, there is no known matter that behaves that way.
How one obtains negative mass is not part of the solution, and it is not currently thought that it is possible. All known sources of matter or energy have non negative energy density. The negative mass energy makes the space contract in front of a presumed spaceship, and expand back to normal behind it. Thus as it travels the space in front it can cover less space, which then expands behind it to be much more space, and thus travel much faster, at so called warp speeds. But the local speed would always be less than c, just globally, after the expansion of space behind the spaceship, it would represent an equivalent faster than light travel.
Now, it is fair to mention that we still don't know what quantum gravity will say, and whether at high enough energies things would be different, and all stated above will not be true. But we do know that that is not likely to enter in until we get to energies on the order of the Planck energy, many orders of magnitude from where we are now. Only black holes and the earlier instant of the universe's Big Bang have achieved those energies.
You can see what might happen with 'effective faster than light travel' in the latest Star Wars movie, and probably most recent science fiction. Emphasis on fiction
Undiscovered negative mass/energy density matter, meaning a lower mass/energy density than vacuum, with another name being exotic matter, is speculative. Alcubierre and others have argued that one could achieve the same with a transversable wormhole. Wormholes, connecting two regions of spacetime, have also been found as solutions of General Relativity's equations. It turns out that they are not transversable (their holes or necks close faster than you can go through them), unless they are made up form exotic matter also.
So the physics of it is speculative.
The effect it would have on human evolution and its future is also clearly speculative, but if we had that we'd be able to do interstellar and intergalactic travel in much shorter times, presumably like we travel on airplanes or so today. We'd either learn how to terraform other planets or it'd be semi useless, maybe we'd find a way to find just the earth-like planets at the same states of its geological and atmospheric state that earth is in now (humans have existed for about 50000 years on earth, and the sun will get hot enough that in about 500 million years the average temperature on earth would go from about 60F to about 120F, unless we make it happen much faster or block solar energy, so a planet may be earth-like for some period of time only, maybe a couple billion years). Surely we'd need a source of almost endless energy (like controlled fusion or ?) to muster the resources to do any of that.
On physics, hard to tell what exotic matter would be or do. And hard to tell if there are other 'exotic' ways of going faster than light globally, clearly nobody knows.
But if we did, we'd be not only stardust, but also could be all over the universe. If we don't kill each other first of course.
And we could run into other intelligent species, almost surely there are others, and then be like Star Wars, or ....