How the Alcubierre Drive Works
Normally, Einstein's theory of relativity doesn't permit any object to travel faster than the speed of light, because accelerating up to that speed requires an infinite amount of energy. The Alcubierre drive gets around this by proposing that the drive would actually manipulate spacetime itself, causing the space in front of it to contract while the space behind it expands. This "warp bubble" allows the ship to reach a destination faster than a light beam traveling through "normal" spacetime.
According to relativity, space is malleable, which is how the Alcubierre drive achieves this feat. (The early universe, for example, expanded faster than the speed of light because spacetime itself can expand faster, even though objects within spacetime cannot accelerate faster.) In this scenario, the ship containing the Alcubierre drive actually sits still and is carried along the warp bubble, kind of like a surfboard riding on an expanding wave. This means that time dilation and other relativistic effects aren't significant, despite the intense speed.