Time for some star trek nerddom to correct stuff in this thread
The warp drive doesn't fold space or make the universe move relative to you. It creates a bubble of normal space around the ship, then makes that bubble go faster than light by shaping the bubble so that the space surrounding the bubble squeezes on it - like what happens when you squeeze a watermelon seed between your fingers. It doesn't violate any known laws of physics (such as the light speed barrier) since the ship actually isn't moving within its bubble, and therefore you don't have any matter actually going faster than light.
It's as though you have a Yugo capable of 30mph. If you put it in a container truck capable of 60mph, the container truck doesn't suddenly find itself unable to go 60 because of the yugo's speed limit of 30. Similarly, the yugo finds itself going twice as fast as it normally could.
We're taking the ship, with the speed limit being that of matter (light speed), and putting it in a container truck that does not have that speed limit. The ship therefore can now travel faster than it should be able to.
Will it work in the real world? Well all I'll say for that is that, when Stephen Hawking was on the set of Star Trek: TNG for a cameo role, he looked at the warp engine in the engineering set and said "I'm working on that."