> We're dealing with gravitational waves here, which travel... on... space-time itself?

It's probably easiest to get there by considering the breaking of symmetries of spacetime.

Let's start with the maximally symmetric spacetime: Minkowski spacetime AKA flat spacetime. We can choose any point in the entire spacetime and measure the same field values everywhere. Conventionally we'd normalize the everywhere-and-everywhen-identical measurements to zero.

Let's add a spherically symmetric central mass of uniform density, giving us the Schwarzschild spacetime. This spacetime has two new properties: firstly, there is a location-dependence of the field-values, and secondly the spacetime is asymptotically flat. The gravitational field values will depend on spatial distance from the central mass, and conventionally we normalize so that they will be larger closer to it and smaller further from it. If our central mass emits light, we get the same result: a falling-off of the local intensity of the light with distance, and the "light field" is just the value everywhere of the intensity of the light. At great enough distance, the measurement of gravitational field values is practically indistinguishable from what one would get if measuring Minkowski spacetime anywhere. Likewise, if the object is bright, then at great enough distance we fail to gather up enough light to see the central object: in every direction we look, we see the blackness expected of an empty spacetime. Our field values depend on one particular space-like coordinate; we can vary the other two, or the time-like coordinate, and the field values remain unchanged.

Let's now add a bump on the surface of the sphere, much like a large mountain made up of the same uniformly dense material. Now our gravitational field values will depend on where we measure relative to the bump. The field values at a pair of adjacent points in this spacetime will differ from those we would get in Schwarzchild spacetime. For example, if we see the bump as on the left of the sphere-with-bump then if we measure slightly clockwise or counterclockwise around the object we will see a change in angle reported by a sensitive mass detector that we would not see in the case of the perfectly smooth sphere. If our bump is much brighter than the sphere, and they are both mutually transparent, then we can also tell that the bump is on the left, or behind, or in front of the sphere on which it rests.

This is still an asymptotically flat spacetime: at great distances, our measurements barely register the difference from true Minkowski spacetime. The source of light and gravitation is pointlike at large but closer-in distances so is asymptotically Schwarzschild. Closer in still we begin to register observables that betray the presence of the bump. This is still a time-independent spacetime, though. The field values (gravitational or visual) at a given point are identical at all times; they only differ between points where there is a difference in at least one of the space-like coordinates.

Now let's add some angular momentum, and have the bump be on the equator of the spinning sphere. Now if we choose a point in spacetime, and with a fairly tame choice of coordinates (polar, say, or Cartesian, or their 4-d extensions Schwarzschild coordinates or Minkowski coordinates) if we choose a point in the equatorial plane and hold the three space-like coordinates constant then the field value is time-dependent -- it varies with the angular momentum of the rotating sphere-plus-equatorial-bump.

Let's make the bump bright, and the sphere it sits on dim, and both opaque. Visually, in this time-independent spacetime, if we are in the equatorial plane and not too far away, we will see the bump disappearing behind the bulk of the sphere, then reappearing, brightening as it gets closer, reaching a brightness peak when it closest to us, then dimming prior to its eclipse. If the rotation rate is constant, then the field-values at our chosen equatorial-plane point rise and fall according to the rotation rate. If we choose different points anywhere in this spacetime, we can predict what the field-values will be based on straightfoward laws.

One thing to note is that the gravitational and light measurements line up: gravimetry and optical telescopes will agree that the "bump" is visible or not, or is closest to us, or not.

The instant of contact during a neutron-star/black-hole (NS/BH) inspiral corresponds to the bright bump on our dim sphere. Future to that period the spacetime looks more and more like Schwarzschild. Past from that period, the bump resolves into a separate roughly spherically symmetrical body (the NS) in a mutual orbit with the BH, and further past the orbit is wider and slower. So we have a more complicated, dynamical time-dependent spacetime; it's still asymptotically flat (and far out but less far, it's still Schwarzschild-like): at enormous distances one might not be able to resolve the two objects -- you get a pretty stable point-like system with all but the most exquisitely precise (to the point of physically implausibly precise) instruments --and at even greater distances, this spacetime will look like empty gravity-free void.

The real spacetime around an astrophysical NS/BH merger has a lot of other masses, and a metric expansion of space at large distances. However, we can remove some of that "foreground" and adapt to the difference between the expanding background and an asymptotically flat one, and reasonably predict the field-values a given NS/BH merger would create at a point in spacetime corresponding to a laboratory on Earth at a given time.

> waves ... which travel

The distribution of the electromagnetic (tensor) field-values generated by the objects in the paragraphs above obey the massless wave equation. Since the distribution of the curvature (also tensor) field-values generated by them line up everywhere, then they too must obey the massless wave equation.

In practice, we divide up spacetime into time-and-space, and the tensors into time-varying vectors and scalars. We choose an idealized background -- Minkowski or Schwarzschild -- and then "correct" it to match what our less-and-less symmetrical systems generate. At reasonable distances, we can ignore most post-Newtonian corrections, and do perturbation theory on linearized gravity. In that, we take a view of a detector at a point (here on Earth, say, sufficiently far away) gathering up a field value that is the result of some past configuration of the inspiralling-binary source. The detector-result is essentially a deviation from the value one would expect from the background that would be sourced by the inspiralling binary in a much more symmetrical (i.e. Schwarzschild or even Minkowski) configuration. The signal is, importantly, periodic.

The theoretical object which encodes gravitational waves is the difference between an idealized time-independent maximally-symmetrical metric and the dynamical time-dependent metric as inferred from actual measurements, minus all contributions to the real metric other than that of the target binary. That is, a perturbation field. One would write this as g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} + O(\{h_{\mu\nu}}^2) + ...
where g is the real, measurable metric, \eta is the Minkowski metric (although one could use a less symmetrical background if one wished), h is the perturbation field, h^2, h^3, ... are higher-order perturbations caused by general-relativistic effects.
If those higher-order effects are small because the binaries are moving slow compared to light and aren't too far away in the universe, then we can ignore them, and \eta combines with h linearly.

Electromagnetism is a linear theory in this limit too, handily, and there are analogies that can be drawn in this limit. One of those is that gravitational radiation propagates in a wave-like way very similarly to electromagnetic radiation, and further that we can analogize between the electric and magnetic parts of an EM wave and the squash and strain parts or + and x parts of a gravitational wave.

PS: I didn't have time to proof-read this before submitting, and hope I won't have to come back in some hours and apologize for howling errors or crazily-hard-to-follow leaps. :-)