I agree that the explanation is pretty vague. They must have based it on the change of doppler shift in multiple consecutive transmissions (as the plane was flying further south it was flying away from the satellite rather than parallel). It would be a bit more credible if they also showed their estimates for the other transmissions, not just the last one.
Bottom line is that they must have better information and be reasonably confident about it otherwise they wouldn't be concentrating on a single area. Kind of how it didn't make sense to me why they were searching on the wrong side of the Malaysian peninsula until they disclosed the primary radar data.