Gravitational waves are a consequence of the curvature of spacetime itself, rather than a wave propagating through a medium like sound waves or water waves. In empty space, gravitational waves propagate by causing the curvature of spacetime to oscillate or ripple as they pass through.
According to Albert Einstein's theory of general relativity, mass and energy cause spacetime to curve. When a massive object accelerates or undergoes asymmetrical motions, it creates ripples in the fabric of spacetime. These ripples are what we refer to as gravitational waves.
Gravitational waves can be thought of as disturbances in the curvature of spacetime that propagate outward from the source. As they travel through space, they carry energy away with them. However, they do not require a physical medium or matter to propagate because they are the very ripples in spacetime itself.
To give an analogy, consider a stretched elastic sheet representing the fabric of spacetime. If you place a heavy object on the sheet, it creates a dent or curvature in the sheet. Now, if you quickly shake or move the object, the dent will start to oscillate and propagate outward as waves on the sheet, even though the sheet itself is the only thing present. Similarly, gravitational waves are the ripples or oscillations in the curvature of spacetime caused by the motion of massive objects, and they can travel through the vacuum of empty space.
Gravitational waves were first directly detected in 2015 by the Laser Interferometer Gravitational-Wave Observatory (LIGO), confirming Einstein's predictions and opening up a new window into observing the universe.