In my opinion, they can't make sense--I strongly favor the outcome that they made a mistake somewhere. But for the sake of arguments, what *could* this result be saying? There are a lot of loonies out there who are starting to stand up to say, "See, I told you so! I've been saying Einstein was wrong for decades." How long before we hear someone say, "This proves that you have to shut down the LHC!" but I digress.
1. Neutrinos really do travel faster than light
Well, that would be something! The Lorentz Transformations were derived towards the end of the 19th century in order to make sense of the Maxwell Equations of electricity and magnetism. Our whole technological world is built on these equations, so anything that would allow the neutrino to go faster than light would have to be a small (very, very, very small) correction on these robust equations. We would have seen this somewhere else by now, I think.
One way to keep everything in tact is to say that the neutrino has an imaginary mass. (What is an imaginary mass? How much would imaginary mass weigh in a gravity field? Negative weight? I have no idea.) Then according to Lorentz, the faster the neutrino goes (beyond the speed of light), the lower its energy gets. So infinitely fast neutrinos would have no energy. But they would have infinite imaginary mass. Is this dark energy?
2. Neutrinos fake going faster than light
This is the non-mistake outcome that I favor. Let's say that the momentum of the neutrino is very well known. Then the Heisenberg Uncertainty Principle says that the position of the neutrino is uncertain. What if there was some effect that enhanced the neutrino's position uncertainty in its direction of motion? Then it would arrive slightly before it got there all along its direction of motion. Maybe it has to do with the neutrinos' interesting spin signature. We know that there is only one spin-type for the neutrino: left-handed spins for neutrinos and right-handed spin for antineutrinos. Maybe the left-handed spin pushes the uncertainty forward a little bit, and the right-handed spin pushes it back. This is easy (enough) to test at OPERA or here at MINOS.
This is, sort of, a combination of the Star Trek warp drive and the Hitchhiker's Guide infinite improbability drive. I like it.