Gravitational waves are a consequence of the curvature of spacetime, as described by Einstein's theory of general relativity. They do not require a medium or matter to propagate through empty space.
According to general relativity, mass and energy can curve the fabric of spacetime. When massive objects accelerate or undergo asymmetrical changes in their distribution of mass or energy, they create ripples in the fabric of spacetime, which we perceive as gravitational waves.
Gravitational waves are similar to ripples on the surface of a pond caused by a stone thrown into it. The ripples propagate outward, carrying energy away from the source. Similarly, gravitational waves propagate through space, carrying energy from the source of the gravitational disturbance.
However, unlike waves in a medium like water or sound waves that require a material medium to travel through, gravitational waves are disturbances in the very fabric of spacetime itself. They travel through empty space without the need for any matter or medium to carry them.
Gravitational waves are incredibly weak and challenging to detect because they interact very weakly with matter. It took until 2015 for scientists to directly observe gravitational waves using extremely sensitive instruments called interferometers, which can detect tiny changes in the lengths of their arms caused by passing gravitational waves.
So, in summary, gravitational waves propagate through empty space by causing ripples in the fabric of spacetime itself. They do not require matter to carry them, as they are a consequence of the curvature of spacetime caused by mass and energy.