= Solution
The starting point is a physical massless closed-string state
$$
\epsilon_{\mu\nu}\alpha_{-1}^\mu\widetilde\alpha_{-1}^\nu|k\rangle,\qquad k^2=0.
$$
Its symmetric traceless part is a <spacetime> spin-two field, the <graviton>; the antisymmetric part is the <Kalb–Ramond field>, and the <scalar field> part gives the <dilaton>. The physical-state conditions impose <transverse polarizations>, and longitudinal null states identify
$$
\epsilon_{\mu\nu}\sim\epsilon_{\mu\nu}+k_\mu\xi_\nu+k_\nu\xi_\mu
$$
for the symmetric component. In position space this is the linearized diffeomorphism $\delta h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$. Thus the massless spin-two state has the <gauge redundancy> required of a gravitational field, rather than an arbitrary massive tensor interaction.
Closed-string amplitudes factorize on this <graviton> pole. Its leading soft coupling is universal and couples to the <energy-momentum tensor> of the other string states. Decoupling <longitudinal polarizations> enforces the gravitational <Ward identities>. The two-derivative cubic <graviton> vertex extracted from the low-momentum three-point amplitude is the cubic vertex of the <Einstein-Hilbert action>; continuing these <Ward identities> requires the nonlinear diffeomorphism-invariant completion. A complementary way to derive the same dynamics is to demand quantum consistency of propagation in a curved background.
Replace the flat target metric in the <worldsheet> action by $G_{\mu\nu}(X)$, and allow the two-form and <dilaton> backgrounds. After Euclidean continuation, their couplings have the schematic normalization
$$
S_\sigma=\frac1{4\pi\alpha'}\int d^2\sigma\,\left(\sqrt h\,h^{ab}G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu+i\varepsilon^{ab}B_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu+\alpha'\sqrt h\,\Phi(X)R^{(2)}\right).
$$
These backgrounds are couplings of a <string nonlinear sigma model>. Quantum <Weyl invariance> requires their Weyl anomaly coefficients to vanish. To first order in $\alpha'$, the metric equation is
$$
0=\beta^G_{\mu\nu}=\alpha'\left(R_{\mu\nu}+2\nabla_\mu\nabla_\nu\Phi-\frac14H_{\mu\rho\sigma}H_\nu{}^{\rho\sigma}\right)+O(\alpha'^2),\qquad H=dB.
$$
The <dilaton> and two-form equations accompany it. In a critical-dimensional background with $H=0$ and constant $\Phi$, the leading metric equation is $R_{\mu\nu}=0$, precisely the vacuum <Einstein field equations>. Matter and other massless string fields supply the corresponding stress-energy when retained. This argument also explains why arbitrary curved target metrics are not consistent string backgrounds.
The equations arise from the leading <string-frame massless effective action>
$$
S_{\mathrm{eff}}=\frac1{2\kappa^2}\int d^Dx\,\sqrt{-G}\,e^{-2\Phi}\left(R[G]+4(\nabla\Phi)^2-\frac1{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+O(\alpha')\right).
$$
A constant-dilaton expansion of its curvature term directly gives the Einstein-Hilbert kinetic and self-interaction terms. For a varying <dilaton>, introduce the <Einstein-frame metric> $G^E_{\mu\nu}=e^{-4\Phi/(D-2)}G_{\mu\nu}$. The <Weyl rescaling> of the metric and the corresponding transformation of the <Ricci scalar> and an <integration by parts> give
$$
S_{\mathrm{eff}}=\frac1{2\kappa^2}\int d^Dx\,\sqrt{-G_E}\left(R[G_E]-\frac4{D-2}(\nabla_E\Phi)^2-\frac1{12}e^{-8\Phi/(D-2)}H_E^2+O(\alpha')\right).
$$
This displays ordinary Einstein gravity coupled to the <dilaton> and two-form. Gravity need not be the only low-energy field: those fields, compactification moduli and <gauge fields> have to be retained or stabilized as appropriate. Four-dimensional gravity additionally requires compactification from the critical target dimension, with its Newton constant depending on the compactification volume.
The approximation is controlled when external energies obey $E\sqrt{\alpha'}\ll1$ and curvatures obey $\alpha'|R_{\mu\nu\rho\sigma}|\ll1$. Massive string modes then produce suppressed higher-derivative interactions, such as higher-curvature corrections. Small <string coupling> $g_s=e^{\Phi_0}$ also suppresses higher-genus quantum corrections; dimensionally $G_D$ scales as $g_s^2\alpha'^{(D-2)/2}$. Thus \b[Einstein gravity is the universal leading two-derivative dynamics of the massless closed-string spin-two field], with additional massless matter, string-length corrections and quantum corrections systematically controlled.
For the bosonic theory this is the massless-sector effective description in 26 dimensions; the <tachyon> means that it is not by itself a proof of a stable vacuum. Tachyon-free supersymmetric <string theories> have the same massless <graviton> mechanism in ten dimensions before compactification. Removing the <tachyon> and obtaining a realistic four-dimensional model require extra physical input beyond the emergence of the Einstein action.
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