What does it mean when we say, "Gravity is a source of gravity"?

You can see this in many different ways. Here is one explanation:

The field equations of electrodynamics in harmonic gauge reads \begin{align} \Box A_\mu=J_\mu \end{align} This is a linear theory and the solution can be determined in terms of the sources, i.e. \begin{align} A_\mu(x)=\int d^3x' G(x,x')J_\mu(x') \end{align} where $G$ is the appropriate Green's function.

You can similarly formulate Einstein gravity in harmonic gauge. If you define $h^{\mu\nu}=\sqrt{-g}g^{\mu\nu}-\eta^{\mu\nu}$, then the Einstein equations read \begin{align} \Box h^{\mu\nu}=T^{\mu\nu}+\Lambda^{\mu\nu}(h) \end{align} where $T^{\mu\nu}$ is the matter stress tensor (analog of $J$) while the second term $\Lambda^{\mu\nu}$ is a complicated expression at least quadratic in $h^{\mu\nu}$. Therefore the solution is \begin{align} h^{\mu\nu}=\int d^3x' G(x,x')\big(T^{\mu\nu}(x')+\Lambda^{\mu\nu}(x')\big) \end{align} Therefore you see that $\Lambda$ which is due to the spacetime curvature, acts as a source to $h^{\mu\nu}$, which encodes the curvature. That is the curvature sources itself. For more details, see e.g. Poisson and Will, section 6.

This is a generic property of nonlinear theories. Quantum mechanically, it means that photons do not interact with each other, but gravitons do.

You can correctly answer either yes or no to the question of whether gravity is a source of gravity. If you want to say no, then you can say that gravitational fields don't contribute to the stress-energy tensor, and that this has to be so because of the equivalence principle -- you can't make a tensor out of the gravitational field, because it vanishes for a free-falling observer, and a tensor that is zero in one set of coordinates is zero in all coordinates. On the other hand, you can say that the answer is yes, but that the gravitational field doesn't appear as a source in the Einstein field equations only because the EFE are too local in scope. On this interpretation, you can say that the gravitational energy exists, but just can't be localized.

As an example, you can say that the external gravitational field of a black hole arises either from the energy of its gravitational field or from the matter that fell in. On a Penrose diagram, you can draw a series of Cauchy surfaces, and every infalling particle's world-line intersects every one of these surfaces, so you would say that the source of the field is always that matter. But this is sort of tautological, because the definition of a Cauchy surface requires that it intersect the world-line of every particle. You can also draw a surface of simultaneity that intersects the singularity, and perhaps intersects absolutely none of the infalling matter, in which case you would say that the external gravitational field that exists now is 100% created by the equivalent mass of the gravitational field of the black hole.

In the case of the planet earth, when it formed by gravitational collapse there was a lot of gravitational energy that was converted into heat. During the whole process of collapse, the distant field was exactly constant. In this example, it's hard to avoid the conclusion that the distant gravitational field can be equally well sourced by either heat energy or gravitational energy.

The reason that it's not straightforward to pass from the local to the global is that you can't integrate a vector over a surface in GR due to the ambiguity of parallel transport. This is different from electromagnetism, where you can prove the global form of Gauss's law just by integrating the divergence.