"As they get closer and closer to one another, they cannot escape each other's gravity, and end up smashing into each other, sending gravitational waves rippling through spacetime.'
In classical physics the two masses would continue circling each other indefinitely, but Einstein showed that accelerating a mass creates those spacetime waves, which carry energy. Thus if two masses are bound in a mutual orbit they are considered be "accelerating" constantly (I think), and the resulting waves bleed off the system's angular momentum, transferring it to the universe as a whole.
Normally such losses are minuscule even over billions of years, but not when many millions of solar masses are involved. Still, the waves are too weak to detect from this distance, altho they don't stay that way.
As the holes draw closer they orbit faster too. That, and the tighter orbital radius, promotes bigger and better emission of gravity waves, speeding up the closing rate more and more. It's an exponential increase, so toward the end things happen fast. The frequency/power of the waves begins to rapidly go to the max (a lot), and then even a very weak force like gravity will make the universe sit up and take notice.
During the last days, hours, or seconds before merger, colossal space warps are emitted at a furious rate. It may be these end-stage-merger waves that are being directly detected or maybe those from the actual merger, not sure.
Don't ask me what happens at the merger, can't handle the math. ;-/