But black-hole pairs orbiting one another emit something else: gravitational waves. These ripples spread outward like sound waves, ever so slightly stretching and squashing spacetime itself as they pass. The first ever to be caught by Earthly detectors, in September, proved Albert Einstein, who predicted their existence a century earlier, was right.
General relativity is a classical field theory. In such theories, disturbances in the fields propagate as waves. The basic equation of GR is the Einstein field equation, which is a wave equation.
GR replaced Newtonian gravity, which was a theory in which gravity acted instantaneously at a distance. Even as early as 1905, long before GR, it would have been clear that the Newtonian picture was inconsistent with relativity, which doesn't allow cause and effect to propagate at speeds greater than $c$.
There is a funny twist in the story, which is that in 1936, Einstein and Rosen published a paper claiming that gravitational waves did not actually exist. The paper was seen to be incorrect by a referee, but Einstein an Rosen did not initially accept the criticism.
Predicted in 1916 by Albert Einstein on the basis of his theory of general relativity,
Assuming you understand general relativity, why not consult the papers given there?
(2) Einstein, A (June 1916). "Näherungsweise Integration der Feldgleichungen der Gravitation". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 688–696.
(3) Einstein, A (1918). "Über Gravitationswellen". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 154–167.