The Green-Schwarz action is a natural supersymmetric extension of the Polyakov action (with a $B$-field which I will omit in what follows since it is not relevant to the question).

For a morphism $X:\Sigma\to M$ from a worldsheet $X$ endowed with a (pseudo-)Riemannian metric $h$ to a stacetime $M$ endowed with a (pseudo-)Riemannian metric $g$, the Polyakov action is usually written in Physics textbooks as $$ S_P[X,h]=\int_\Sigma d\sigma^2 \sqrt{h} h^{ab}\partial_aX^\mu\partial_bX^\nu g_{\mu\nu} $$ It is very simple to rewrite this in a completely intrinsic way. Namely, the differential of the smooth map $X$ is a morphism $dX:T\Sigma\to X^*TM$ of vector bundles over $\Sigma$, while the metric $h$ induces a canonical volume form $dvol_\Sigma$ and a cometric $\gamma_h$, which is a section of $T\Sigma\otimes T\Sigma$. Finally, $X^*g$ is a morphism $X^*(TM)\otimes X^*(TM)\to \mathbb{R}_\Sigma$ and the Polyakov action is intrinsically given by $$ S_P[X,h]=\int_\Sigma dvol_\Sigma \, X^*g((dX\otimes dX)(\gamma_h)) $$ Equivalently, and possibly more nicely, one can rewrite this in terms of the Hodge star operator $\star_h$ for the metric $h$ as $$ S_P[X,h]=\int_\Sigma X^*g(dX\wedge\star_h dX) $$ The differential $dX$, as an element in $\Omega^1(\Sigma;X^*TM)$ can actually be seen as the pullback via $X$ of the canonical 1-form $E$ in $\Omega^1(M,TM)$ corresponding to the identity morphism of $TM$ over $M$. That is, $dX=X^*(E)$ and the action becomes $$ S_P[X,h]=\int_\Sigma dvol_\Sigma \, X^*(g(E\otimes E))(\gamma_h)= \int_\Sigma X^*g(X^*(E)\wedge\star_h X^*(E)) $$ Written this way, the action is immediately translated in what physicists call the vielbein formalism: if $\{e_1,e_2,\dots\}$ is a local orthonormal frame for $TM$ then $X^*E$ is written locally as $E^\mu_\nu e_\mu dX^\nu$ so that if one adopts the shorthand notation $E^\mu_a$ for $E^\mu_\nu \partial_a X^\mu$ the action reads $$ S_P[X,h]=\int_\Sigma d\sigma^2 \sqrt{h} h^{ab}E_a^\mu E_b^\nu \eta_{\mu\nu} $$ which is another of the forms one often finds the action written in textbooks.

Then the action is promoted to its supersymmetric version simply by replacing the vielbein $E_a^\mu$ with the "supervielbein" $\tilde{E}_a^\mu$ (whatever it is: this is something I have not clearly understood, yet).

My question is: what is the supergeometry behind this?

I would expect one has some supermanifold $\tilde{\Sigma}$ with underlying classical manifold $\Sigma$ and $\tilde{M}$ with underlying classical manifold $M$, as well as a map $\tilde{X}: \tilde{\Sigma}\to \tilde{M}$ coming into play, together with supermetrics $\tilde{h}$ and $\tilde{g}$, but I'm unable to find a reference presenting the action directly in these geometric terms, nor to guess exactly which the precise supergeometrical picture should be from the supervielbein description of the action (the only thing I'm reasonably sure about is that $\tilde{M}$ should be the supermanifold associated with some spinor bundle over $M$).