The "carrier" is the metric tensor. Roughly, at every infinitesimal point in a spacetime like ours there is a tensor value which encodes the distance from that point to its neighbours along all four orthogonal dimensions. "The metric" of a spacetime is usually taken to mean an integration of the metric tensor values at each point along some set of points.

Let's draw a supermassive black hole binary. In the rough <https://en.wikipedia.org/wiki/Minkowski_diagram> schematics below, each equal-mass black hole is an O and we show an observer ("y" for "you") who sees the binary's circular orbit edge-on. The y axis is time, oriented with the future towards the start of this comment. The x axis is one spatial dimension. The diagram-angle from the binary to the observer is meant to be 45 degrees, representing a null aka lightlike separation.

Let's consider two snapshots of the binary's mutual orbit:

1.

y

OO

2.

y

8

The "OO" vs "8" is an abuse of notation; read the OO as the orientation where "y" sees one eclipsing the other, and the 8 as the orientation where y sees the two distinctly.

"y" is at a spatial remove. But "y" is drawn a couple of lines up because of the finite propagation speed c. The orientation OO...y or 8...y is not felt by y at the time, but later. A null curve connects the binary "at the time" with the "but it's felt later at" observer.

We then integrate all the infinitesimal points along the null curve between y and the centre of mass ("COM") of the binary and find that the null curve in diagram 1 is shorter* than the null curve in diagram 2, because there is more mass between y and the COM in the first diagram.

Above, "y" is pointlike and y would free-fall towards the centre of mass. In y's proper time the free-fall would be faster during orientation 1 than during orientation 2. Since the binary is in a mutual orbit, there is a speeding-up and slowing-down of the free-fall measured by y in y's proper time.

Now we consider some extended-body stuff. Things above and below the SMBHB orbital plane will tend to fall towards the plane. If "y" has some height and the binary isn't treated as practically pointlike, the various parts of now line-like y want to free-fall to the black hole closest to them. In orientation OO the top and bottom of the y will squash inwards. In orientation 8 the top and bottom of the y will stretch up and down. Interpolate through the orientations between OO and 8. The "height" here is along one axis in the plane perpendicular to the binary's orbital plane; a "y" with "width" would be extended in that plane in the orthogonal axis. In orientation 8 there is a stretch along that axis compared to OO.

The magic is all in the orientation of the observer/gravitational-wave-detector to the two mutually-orbiting bodies, and the integration of the metric tensor between every part of the observer and every part of the binary. This is ... generally analyitically intractable, so one does numerical methods ("NR", numerical relativity) or one works in the approximation known as linearized gravity. Most non-specialists, and even many members of the LIGO/Virgo/KAGRA et al collaborations who have encountered the mathematics of gravitational waves did so in the context of linearized gravity, and only sometimes encounter NR and (assuming they aren't writing the NR codes) even then don't think too hard about how the "block universe" spacetime described by the field equations are split into 1+3 dimensions.

So, in summary, the crucial thing is the length of the null curves connecting every part of the y with every part of the binary. The periodic rotation of the binary causes the lengths and angles of those curves to oscillate, shortening and spreading or lengthening and narrowing. Looking closely at a large number of such null curves at once, one can successfully model them in bulk as obeying the massless wave equation, even though nothing actually propagates (the whole spacetime-filling metric tensor field is solved all at once; it only "evolves" or "propagates" when we think in terms of the initial values formalism or a 1+3 formalism).

Finally, here is an pulsar-timing-array astrophysicist (<https://www.aoc.nrao.edu/~tcohen/>) doing an intepretive dance (after a bit of explanation):

https://www.youtube.com/watch?v=uH91gSI4ELs&t=0s

(there are further links in the video description).

- --

* someone is bound to pounce on "shorter" given that the interval of a null curve is always 0 (thus the name and from that flows crazy wrong ideas about "photons experience no time"), but we can use the very much timelike worldlines of the (components of the) binary and observer and retarded time to construct a notion of the length of the null curves connecting them. Eric Poisson has gory technical details for anyone actually interested, in the context of the self-force formalism which is more suitable than linearized gravity for high-mass-ratio binaries, especially as the binary hardens.

PS. "Fabric of spacetime" is a cliché that causes me to grit my teeth. I grit harder when someone who has worked with the Einstein Field Equations writes or says it. It's not only overused, it's misleading to non-experts (and sometimes even to experts). Spacetime is not a substance. It does not "distort", a solution to the EFEs is what it is; "distortions" imply comparing the solution to some alternative that one likes better (e.g. the spacetime without any matter, despite any non-zero stress-energy in the solution). It is at best a mathematical container for coincidences ("events" where two objects at the same point in spacetime can have e.g. their velocities usefully compared) and interactions like collisions, scatterings or the formation of molecular bonds.

Unfortunately, colloquially or informally it's very hard to talk about gravitational waves without talking about the stretching and squashing of some region of space (not spacetime), because people (even experts) internalize "fabric" and similar metaphors, because formalisms and approximations to general relativity used in studying inspiralling-binary sources of gravitational waves split spacetime into space + time, and don't invite the consideration of solutions of the geodesic equation (for tractability reasons).