In open space, remote from any masses: there is only uniform ticking. Alice notices all the clocks ticking at the same rate: represent this space-time structure by aligning all the light cones.
Alice notices that all the cones at Bob’s height tick more slowly than at her greater height and that the clocks at Charlie’s height, closer still to the mass, tick more slowly still. The space-time structure varies near a mass: the clocks tick more slowly, so the light cones, showing the local causal structure, are tilted by more and more.
As a gravitational wave passes in deep space, so the cones vibrate: space and time are changed as the wave passes. The gravitational wave is a wave in the structure of space and time.