Anything we do daily will "rewire our brains" ... unfortunately, nothing practical can be done with these results, and scientists still really don't know how to interpret them. So, I believe that fMRI studies such as this one really only serve as curios ...
As a first guess, which the researchers themselves offer, the marker of internet savvy that they found was the frontal lobe processing, which is roughly that used in decision making, synthesis, and discrimination, is invoked. In other words, during internet search, savvy users constantly assessing the credibility of sources and validity of answers. This gives directly two principles which Google has used to dominate the market: authority, as encoded in page rank, and the swift checking of relevance and credibility, as helped by the short blurbs of relevant text, the back-links to similar pages or pages that link to it, and the quick viewing enabled by Google cache and 'view as html.' Anything further improving these processes would likely be a significant improvement in user experience.
It's a strength of the scientific method that one can offer a hypothesis without believing it. In any case, there are additional cognitive tests which one can use to examine the actual behavior from a different angle.
>> So, as you become more comfortable with the Internet, you actually use more of the brain. <<
Ok, this sounds like an interesting observation (although I wouldn't call that counter-intuitive). What else did they discover? Couldn't find much more there...
How is this study using fMRI as a scalpel? I think the resolution is more than good enough to isolate activity in decision areas of the brain, which is exactly what they're doing.