The way I see it, there are three reasons it's such a big deal.
The first, as you say, is the (further) confirmation of Einstein's theory of gravity. Newtonian gravity doesn't have gravitational waves. Their existence was already quite established by Hulse and Taylor's discovery and analysis of the binary pulsar PSR B1913+16. The system's orbit is decaying almost exactly as predicted by General Relativity. (Note that in the classic figure, the curve is not a fit: that is the prediction!) The direct detection of gravitational waves, however, better confirms that the waves behave as we expect they would, in the sense that the observations match the so-called "chirp" that we expected.
The second reason is because this is now a new way of measuring things in the Universe. Note that the announcement told us (roughly) the masses of the two black holes. It's actually very difficult to weigh black holes! For example, this 2010 paper discusses constraints on 23 stellar mass black holes, all from the fact that they're in binary systems, and the motion of the partner can be observed. Even then those are mostly mass functions that still depend on the inclination of the orbits, which is often unknown. So the gravitational waves give us a nice measurement, not just confirmation that the phenomenon exists.
With the method proven, there are some fascinating potential applications. One of my favourites is that LIGO (or similar) could potentially detect the formation of a black hole in a core-collapse supernova. Just like the detection of neutrinos from SN1987A confirmed some of our basic ideas about these events, detecting the gravitational waves from the birth of a black hole would inform our understanding further still. And it might even tell us, for example, how massive the black hole is at birth. There's no end to the novel ways that our theories could be tested!
The final reason is simply the astonishing technical achievement. They measured motions of at most a few thousandths of a femtometer over the 4km arms. That's mind-bogglingly precise. Hopefully this will give more impetus to other gravitational wave detectors. Here's hoping that eLISA will fly!
The parallel with the Higgs boson is quite apt. In both cases, we built a big machine to find something that we were pretty sure should be there. (Though the LHC was a tad more expensive...) Now it's a case of using the new tool to observe the Universe, and seeing if the waves behave as we think they should. And who knows, maybe some completely unexpected signal will appear to confound us completely.