For me, it's mainly interesting because two different fields of physics meet, nuclear physics (electron scattering) experiments, and atomic physics (spectroscopy).
Otherwise, it's the same as with all basic science. We don't know what it is "good for" until somebody figures out how to cure cancer with it.
The paper this article refers to is in particular interesting, because it finds a value not in agreement with earlier measurements of the /same/ type. The indicated 3.3 sigma shift in the Rydberg constant, one of the most exact measured quantities in existence, is a little bit worrying, but such shifts happen more often than they should.