The following facts are what I think I know about gravitational waves:
What I still don't know is what are they made of? Are they empty?
A wave is a traveling distortion. This goes for any type of wave. An ocean wave is a distortion of the water surface. A sound wave is a distortion in air pressure. A light wave is a distortion in electromagnetic fields. A wave is made of the thing that is vibrating--ocean waves are made of water, etc. So, a gravitational wave is made of space and time, since gravity is the effect of space and time warping due to nearby masses.
In General Relativity the effect of gravitation is represented by the stress energy tensor, $T_{μν}$:
$$ R_{μν} - \frac{1}{2}Rg_{μν} + \Lambda g_{μν} = \frac{8\pi G}{c^4}T_{μν} $$
where $R_{μν}$ is the Ricci curvature tensor, $R$ is the scalar curvature, $g_{μν}$ is the metric tensor, $Λ$ is the cosmological constant, $G$ is Newton's gravitational constant, $c$ is the speed of light in vacuum, and $T_{μν}$ is the stress–energy tensor.
Gravitational waves are the sinusoidal changes on space time induced by specific time dependent changes in the stress-energy tensor. In a Newtonian frame it is called acceleration , which is not enough to get a gravitational wave, only asymmetric mass distributions will give gravitational waves.
So it is sinusoidal changes in the stress-energy tensor which distort the space time solutions which fulfill the equation of general relativity.
The waves in general are fluctuations, or ripples in something. Gravitational waves are ripples in space(time).
Now, what is space? Space itself is not defined. What is defined in GR is its certain properties. These properties/attributes include things like tensors, metrics, and manifolds. As space is defined only in terms of these properties, a ripple in space has to be a ripple in value of one or more of these properties. Meaning the value of one or more of these properties is rippling/changing. The ripple in the value is propagating at the speed of light.
So, the gravitational waves are made of fluctuation in the value of certain property of space. I am not certain about role of these terms. But per anna v in her answer, looks like the rippling property is the "stress energy tensor".
In common language, the Stretchiness of space is changing, and that change is propagating at light speed. Because of changed Stretchiness, the behavior of space on its path changes slightly.
I may suspect that you may have not grasped the concept of General Relativity very well, but that doesn't matter. The beauty of science is in its explanations.
Imagine you dropping a pebble into a water. You can see waves forming from the distortion in the water. Now try to drop two pebbles into the water close by and see the waves formed by each distortion.
(source: compassblogs.org)
Now as an analogy to the GW, imagine you dropping pebbles continuously close to each other.
Now imagine that you are a 4-dimensional creature and you can see space time in a 4-dimensional view. You will see the waves like this.
The GWs are basically ripples/distortions in the medium that we live in, space-time itself.
That's the simplest explanation that I could give, well some are not accurate but it's to simplify the concept.
Gravitational waves are fluctuations in spacetime, but what is spacetime? Your intuitive sense of space and time tells you what space and time ARE. In science, and physics, in particular, we use words in very specific ways. In physics, spacetime is a combination of distance and time. So, when we say that gravitational waves are fluctuations in spacetime, we are saying fluctuations in distances and times.
Gravitational waves are a prediction of general relativity (GR), a theory of space and time (spacetime), and matter and energy. We talk about observers in GR, and ask questions like, "What would an observer observe when a gravitational wave passes?" Now, you say, "The observer observes fluctuations in spacetime, distance and time, of course." And what does that mean?
It means that the distances between points increase and then decrease and then increase and then decrease, etc. The increases and decreases are above and below, respectively, the distances before and after the wave passes. The same is true for the time between events, like the time between ticks of a clock. As the gravitational passes the observer, the observer's clock runs faster than normal, then slower than normal, then faster than normal, etc.
And what is it spacetime instead of space and time? That's because in GR, and in special relativity, which is a special case of GR, spacetime are unified. What does that mean? It means that observers at different positions and with different velocities relative to each other measure distances and times. One observer may see two events happening simulataneously at two different locations, while another observer sees the two events happening at the same location at different times. So space and time flow back and forth depending on your velocity.
Consider a magnetic field mediated by photons - the classic field lines originating from one end of a bar magnet to the other.
Now bring in another magnet and 'disturb' that magnetic field.
If you had a meter measuring the energy of the magnetic field it would change when the other magnet was present.
Since gravitons are the theoretical force particles that mediate gravity across distance just like photons mediate electromagnetism across distance, we can use the same analogy.
Here on Earth we have many sources of gravitons, the Sun, Earth, Moon, etc., so we are living in an ocean of gravitons. As you said, when two black holes collide they create a disturbance in their local gravitational field that then travels across the universe within the field of gravitons that already exist. When that wave enters our solar system and changes the potential energy of our local graviton field, we can detect that change.
So, to me, the bottom line answer to your question is, gravity waves are changes in the energy of the local gravitational field of gravitons. In this case, the detectors were set up in such a way as to rule out other changes in the field such as the moon's orbit and trucks bumping over the parking lot.
There are two types of Gravitational Waves (GWs), the first type is "made of" steady state EM fields (normal gravity); the second type is made by dynamic EM fields (large perturbations of the steady state EM fields, depicted in Dendi's answer).
Your stated facts are true, when/if applied to the second type of GW.
