Why is the detection of gravitational waves so significant?

Gravitational waves are qualitatively different from other detections.

As much as we have tested GR before, it's still reassuring to find a completely different test that works just as well. The most notable tests so far have been the shifting of Mercury's orbit, the correct deflection of light by massive objects, and the redshifting of light moving against gravity. In these cases, spacetime is taken to be static (unchanging in time, with no time-space cross terms in the metric). Gravitational waves, on the other hand, involve a time-varying spacetime.

Gravitational waves provide a probe of strong-field gravity.

The tests so far have all been done in weak situations, where you have to measure things pretty closely to see the difference between GR and Newtonian gravity. While gravitational waves themselves are a prediction of linearized gravity and are the very essence of small perturbations, their sources are going to be very extreme environments -- merging black holes, exploding stars, etc. Now a lot of things can go wrong between our models of these extreme phenomena and our recording of a gravitational wave signal, but if the signal agrees with our predictions, that's a sign that not only are we right about the waves themselves, but also about the sources.

Gravitational waves are a new frontier in astrophysics.

This point is often forgotten when we get so distracted with just finding any signal. Finding the first gravitational waves is only the beginning for astronomical observations.

With just two detectors, LIGO for instance cannot pinpoint sources on the sky any better than "somewhere out there, roughly." Eventually, as more detectors come online, the hope is to be able to localize signals better, so we can simultaneously observe electromagnetic counterparts. That is, if the event causing the waves is the merger of two neutron stars, one might expect there to be plenty of light released as well. By combining both types of information, we can gain quite a bit more knowledge about the system.

Gravitational waves are also good at probing the physics at the innermost, most-obscured regions in cataclysmic events. For most explosions in space, all we see now is the afterglow -- the hot, radioactive shell of material left behind -- and we can only infer indirectly what processes were happening at the core. Gravitational waves provide a new way to gain insight in this respect.

Chris' answer provides an excellent explanation as to why gravitational waves are useful to detect in general. Here's my take (as someone who works in the theory of black holes) on what is particularly interesting about the signal that was announced yesterday. Many of my thoughts are taken from the official NSF press conference and from colloquia at my institution.

The Event Itself

Numerical analysis of the gravitational wave event that was measured on September 14, 2015, has revealed a great deal about the nature of the event that took place.

The following is a figure from the LIGO report which shows the gravitational wave signal:

(source)

The red line in each graph is the gravitational wave signal measured from the observatory in Hanford, Washington. The blue line is the gravitational wave signal measured from the observatory in Livingston, Louisiana. The top left graph shows the Hanford signal alone, the top right graph shows the Livingston signal overlaid with the Hanford signal (look how nicely they match up, proving that this was not a local source of noise but rather a signal being generated from some cosmic distance).

The left graph in the second row is most interesting. The light gray line essentially shows the signal, cleared of as much noise as possible (the equipment is so sensitive that all sorts of things can cause slight jitters in the waveform). The red line represents the waveform that would be predicted by the techniques of numerical general relativity for a system of two black holes spiraling into each other. It's no coincidence that the observed waveform (light gray) and predicted waveform (red) overlap so well.

There is, of course, a great deal of analysis that goes into checking the statistical significance of this data. Scientists at LIGO have found that within a statistically significant margin, this waveform was probably produced by a binary system of two black holes, each about thirty times as massive as the size of the sun.

Now, for the specifics as to what is interesting about this event.

Black Holes In General

Before yesterday, we had no direct evidence to show that black holes existed. We were fairly confident in the existence of black holes, but only through indirect measurements. This is the first ever direct measurement of a black hole—the objects in question are massive enough and compact enough that they almost surely must be black holes. What's more, the data fits perfectly our general relativistic predictions as to what kind of radiation will be released by a black hole merger. This is huge news—physicists never had complete evidence that black holes existed before yesterday, although the public might take it for granted. Black holes exist, and they work the way we thought they did. That's incredible!

Types of Black Holes

From an astrophysical perspective, this is quite interesting, because both of the inspiraling black holes were about 30 times as massive as the sun (henceforth referred to as having "30 solar masses"). Astrophysicists had no real compelling evidence for black holes in this mass range. It was assumed that we had black holes in the range of 3-20 solar masses, and the so-called "supermassive" black holes (which are millions, billions, of solar masses? I'm not an astrophysicist so I can't tell you). This is a fascinating astrophysical problem—the mass in a black hole needs to come from somewhere. What is the process by which a black hole of ~30 solar masses forms? From where does it take its matter? How massive is it when it first forms (from a star, perhaps?), and how much does it grow after it has already become a black hole?

Oh, and by the way, we haven't just confirmed the existence of two black holes of solar mass ~30. We've confirmed the existence of one black hole of 62 solar masses—the black hole remaining after the two have merged. Speaking of, let's talk a bit about that final black hole.

Radiation

The collective mass of the two black holes before they merged was ~65 solar masses. The mass of the final black hole was ~62 solar masses.

What that means is that 3 solar masses were radiated away in gravitational waves as the black holes merged. Not impressed? Well, here's some perspective: according to the NSF conference given yesterday, the power output of gravitational radiation during the last moments of the black hole merger was more than the collective power output of every star in the universe combined.

That's a lot of energy, very fast. What happens once that energy is released? Well...

Ring-Down

This is my personal favorite, but it's also the thing about which we have the least information. If you look again at the figure I included earlier in this response, at, say, the second graph in the left column, you'll notice that the pattern goes as follows:

Slight vibrations, increasing in amplitude in frequency, suddenly oscillating very quickly at a high amplitude, and then dying down to almost nothing.

That sudden increase in frequency is called a "chirp," and it's what LIGO was looking for. That chirp tells us everything we need to know about the black hole merger.

But what about what happens afterward? The exponential decay of the signal corresponds to the resulting black hole (with 62 solar masses) settling down into a stable state. The question of black hole stability is incredibly interesting, and the process by which a black hole settles down after some major perturbation (e.g. merging with another black hole) is a fascinating object of study.

Basically, if you hit a black hole, it rings. When you perturb a black hole away from its stable state, you create something called quasinormal modes—mathematical descriptions of the perturbation from equilibrium—which decay exponentially over time as the black hole approaches equilibrium.

The experimental signal does not contain much information about the ring-down. We can't glean much information about exactly how the black hole settles into a stable state—the process doesn't generate very strong gravitational waves, for one thing, and it happens very quickly.

But that's okay. In the figure, we can see it happen. We see two black holes merge, release three solar masses of radiation, and then settle down into a stable final state. That alone is incredibly exciting.

Oh, by the way, one parting thought: this black hole merger happened about a billion years ago. We're only getting its signal now.

In additions to what Chris White lists, I'd like to point to the fact that, except for a few meteorites and some dust collected on the plates of satellites and rocks from Mars (and cosmic rays and a handful of neutrinos; thanks Ruslan and Kyle Oman), until now all information reaching us from the Universe — whether it is the Sun, the more distant planets, other stars, galaxies, CMB, etc, — has come to us in the form of electromagnetic radiation.

Gravitational waves is a whole new mode of gaining knowledge about the Universe. Both from objects where we also see radiation, but also for instance perhaps at some point inflation at the Big Bang, where using electromagnetic radiation we can't see further back than the CMB, 380,000 years after Big Bang (this is what the BICEP2 guys thought they saw two years ago, but it turned out to dust…).