These experiments confirm the classical theory of gravity, which is Einstein’s general relativity, just as Maxwell’s equations are the classical theory of the electromagnetic field. Just as Maxwell’s equations are perfectly adequate for waves of macroscopic intensity, GR is perfectly adequate for astrophysical gravity waves.

A whole separate question is what is the quantum mechanical theory that has general relativity as its classical limit? For electromagnetism, quantum electrodynamics (understood in the 40’s) is the quantized version of Maxwell’s and predicts that electromagnetic energy measurement outcomes come in “chunks” (photons). But, although much is known about features of “quantum gravity” (like that gravitons will be massless, spin 2), there is famously no consensus yet about the precise theory.

As to how to reconcile the force carrier picture with spacetime picture — even classically one can consider an “overall” spacetime background geometry such as that created by the whole earth. Then consider little ripples perturbing this background. Gravitons are these little ripples turned on a quantized amount (heuristically). How the overall background itself gets formed as an “enormous pile of gravitons” will depend on the precise theory of quantum gravity. String theory does have a partial answer to this so can model such things as black holes quantumly.