Their method is a modification of the Alcubierre drive, proposed originally in 1994 by Miguel Alcubierre, which would allow a starship to travel along spacetime. As the spacetime expands in front of the ship and contracts behind it, the ship moves along the "warp bubble" like a surfboard on a wave of water.
With the addition of string theory, the Cleaver and Obousy believe that (in principle) it would be possible to exploit the eleventh dimension from M-Theory (the current permutation of string theory) to create dark energy, which would cause the necessary expansion of space to propel the ship forward. This certainly isn't a cakewalk, though, as explained in the Baylor University press release:
"The Baylor physicists estimate that the amount of energy needed to influence the extra dimensions is equivalent to the entire mass of Jupiter being converted into energy."
Their method is obviously highly speculative and promoted more for public relations effect than because it's practical science. Also, as Peter Woit has pointed out, calling M-Theory new is a bit odd since it was originally proposed in 1995 (by string theory guru Edward Witten). Still, this is the sort of thing which captures the imagination of non-scientists, and helps to turn young people toward a life of pursuing the mysteries of the universe ... so long as we all realize that these sorts of articles do not mean that the Enterprise is right around the corner.
Related Articles:
- Baylor University - 'Star Trek' Warp Speed? Two Baylor Physicists Have a New Idea That Could Make it Happen
- ArXiv.org - Warp Drive: A New Approach
- ArXiv.org - Putting the Warp into Warp Drive
- ArXiv.org - Casimir Energy and Brane Stability


Comments
What about time displacement while you are traveling at warp speed?
There is an error, space-time contracts in front of the ship and expands behind the ship. Not the other way around.
There is another, more energy-efficient possibility: do not change the eleventh dimension itself to create dark matter, just send electromagnetic energy on a returning trajectory through a hidden dimension so that the hidden dimension transforms the electromagnetic energy directly into gravitational itself (instead of the energy-wasting e=mc2 and much matter to create little gravity stuff with dark matter) and then returns the transformed energy (in the form of gravity and antigravity) to nortmal space-time. A way to send electromagnetic energy into a hidden dimension would be to make electromagnetic waves short enough to fit into hidden dimensions. That requires very high-energy photons, but not so many of them since each photon is relatively energy-rich in itself. A solution to the engineering problem would be to focus many laser beams on a nanoscopic amount of matter, heating it to extreme temperatures with modest total energy requirement (preferably iron, the only element that would not go thermonuclear) and measure any disturbance of gravity and loss of electromagnetic energy.