Personally I wouldn't say you're missing something 'trivial' - maybe the result is 'standard' in some way that is taken for granted by people who know the field.
The earliest reference containing the result appears to be Blanchet, Luc, and Thibault Damour. "Post-Newtonian generation of gravitational waves." Annales de l'IHP Physique théorique. Vol. 50. No. 4. 1989 (see equations $2.10a.$ - $c.$ therein).
The entire derivation is quite involved, but the level of complexity depends on what your starting point is.
In the linked paper the result is derived for the near-field, where the post-Newtonian expansion is assumed valid (i.e. $v/c << 1$, and a weak gravitational field). This allows them to make a number of assumptions, including that $\partial_0 g_{\mu\nu}^{in}/{\partial_i g_{\mu\nu}^{in}} = O(\frac{1}{c})$, where $g_{\mu\nu}^{in}$ is the metric in the inner domain where this near-field expansion is used.
If we start with the Einstein equation in harmonic coordinates, the following set of linearised equations can then be derived:
$\Box\ln(-g^{in}_{00}) = \frac{8\pi G}{c^4}(T^{00} + T^{ss}) + O(\frac{1}{c^6})$
$\Box g_{0i}^{in} = \frac{16\pi G}{c^4}T^{0i} + O(\frac{1}{c^5})$
$\Box g_{ij}^{in} = -\frac{8\pi G}{c^4}\delta_{ij}(T^{0i} + T^{ss}) + O(\frac{1}{c^4})$,
where $\Box$ is the d'Alembertian (using the mostly $+$ metric), and the various components of the energy-momentum tensor $T$ are of known order in $c$.
For convenience the authors then define
$\sigma = c^{-2}(T^{00} + T^{ss})$ and
$\sigma_i = c^{-1}T^{0i}$.
This is a clever trick that allows them to simplify the form of the resultant equations
They then define certain potentials (eq. $2.8$ and $2.9$), from which the desired results follow (along with a number of intermediate intuitive leaps to justify the calculations). These potentials themselves satisfy certain linear equations.
I hope this helps, and apologies for any typos. I think the best thing would be to read the linked paper. It is a bit convoluted, and in a sense I think the result you posted is just one possible parametrisation.