By linearizing the metric in the following way (approach in most textbooks):
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\text{ with } |h_{\mu\nu}|\ll 1$
and choosing the transverse-traceless gauge a wave equation for the perturbation $h_{\mu\nu}^{TT}$ is obtained (with $\Box=\eta_{\mu\nu}\partial^{\mu}\partial^{\nu}$:
$\Box h_{\mu\nu}=0$
Therefore possible solutions for the perturbations $h_{\mu\nu}$ are waves with $+$- resp. $\times$-polarization. These waves have an effect on test masses eg. deforming a ring according to the polarization of the wave. So by introducing little springs between the test masses the gravitational wave must carry energy (stored in the springs by friction). But what is the source of this energy?
In Gravitational Waves Vol. I by Michele Maggiore the stress-energy tensor of GWs $T_{\mu\nu}^{\text{GW}}$ is derived by linearizing the metric like this:
$g_{\mu\nu}=g_{\mu\nu}^B+h_{\mu\nu}$
Where $g_{\mu\nu}^B$ is the curved background. This makes sense to me, because GWs themselves are sources of curvature.
I'm confused by the first approach: The GWs seem to carry energy, but in the field equations $T_{\mu\nu}$ is set to zero. So where is this energy coming from? How should the waves be interpreted physically?
EDIT:
After thinking about it again: $T_{\mu\nu}$ is set to zero because we look at a region in spacetime, where there is no matter/energy density and describe how a the perturbation $h_{\mu\nu}$ behaves (like a wave). But the perturbation must be caused somewhere by some source. So the wave is coming from a region with a source (similar to the vacuum wave solution in electrodynamics).
I'm still a bit confused what the physical interpretation of these waves is.