4% sure does seem significant. But more interesting is that the measurement is thought to be much more precise because of the method of measurement. Doesn't it seem more likely that it's just not possible to get an accurate measurement with the electron -- like measuring a grape with a yardstick instead of a micrometer?
Maybe, but this is still surprising. Measure a grape with a metre rule, you should still be able to say 'it's between a centimetre and a centimetre and a half.' Measure it with a micrometer, and you'd expect to see a result like 'It's 1.2144 centimetres.' If the micrometer instead measured the grape at 0.7218 centimetres, well, you'd be puzzled. First of all, of course, you'd check you were doing it right. You'd examine your micrometer and make sure you were operating it correctly. You'd recheck how you measured it with the metre rule - is it zero from where the number is printed, or from the edge of the ruler, is the ruler maybe worn down at the edge?
But if all that checked out and you still had this discrepancy, you'd start to wonder if your ruler and your micrometer were really measuring the same units.
Hence the suggestion of new physics. Theoretically the muon should act like a heavy electron - interacting with the proton in just the same way, so that it can be used as a more precise probe on the size of the proton. It would be the micrometer to the electron's metre rule. If it doesn't - if the muon interacts with the proton in some unexpected way so as to throw the measure off - then we've discovered something beyond the standard model.
There are quite a few indications that there is physics beyond the standard model - heavy neutrinos, the abundance of matter over antimatter, the dark matter - and so if we can add this to the list then maybe it can help pin down just what sort of a new theory we're looking for. We've got to have something to do once the good people of Geneva finally hammer us out a Higgs, after all :-)