The classical theory of spacetime geometry that we call gravity is described at its core by the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy in spacetime.
For example: $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ is an important concept in General Relativity.
Mathematically, how do the Einstein's equations come out of string theory?
(see also: The General Relativity from String Theory Point of View)
$\mbox ds^2=g_{\mu\nu}\mbox dx^{\mu}\mbox dx^{\nu}$ is a definitional fact from Riemannian geometry, and has nothing to do with gravity. The "physics" of General Relativity is contained in the Einstein Field equations $G_{\mu\nu}=8\pi T_{\mu\nu}$, or equivalently the Einstein-Hilbert action $\mathcal{L}=\frac1{16\pi}R$.
Deriving these results from the Polyakov action is hard, but there is a simpler standard approach that you'll find in many textbooks. In string theory, the Dilaton couples to the worldsheet
$$S_\Phi = \frac1{4\pi} \int d^2 \sigma \sqrt{-h} R \Phi(X)$$
The breaking of conformal symmetry in this action can be summarized by 3 functions known as the beta functions. In Type IIB string theory, the beta functions are:
$${\beta _{\mu \nu }}\left( g \right) = \ell _P^2\left( {{R_{\mu \nu }} + 2{\nabla _\mu }{\nabla _\nu }\Phi - {H_{\mu \lambda \kappa }}H_\nu {}^{\lambda \kappa }} \right)$$
$$ {\beta _{\mu \nu }}\left( F \right) = \frac{{\ell _P^2}}{2}{\nabla ^\lambda }{H_{\lambda \mu \nu }} $$
$$ \beta \left( \Phi \right) = \ell _P^2\left( { - \frac{1}{2}{\nabla _\mu }{\nabla _\nu }\Phi + {\nabla _\mu }\Phi {\nabla ^\mu }\Phi - \frac{1}{{24}}{H_{\mu \nu \lambda }}{H^{\mu \nu \lambda }}} \right) $$
Setting these functions to zero (i.e. to require conformal symmetry, expecting to obtain the vacuum Einstein-Field Equations):
$${{R_{\mu \nu }} + 2{\nabla _\mu }{\nabla _\nu }\Phi - {H_{\mu \nu \lambda \kappa }}H_\nu ^{\lambda \kappa }} = 0. $$
$${\nabla ^\lambda }{H_{\lambda \mu \nu }} = 0 . $$
$$ { - \frac{1}{2}{\nabla _\mu }{\nabla _\nu }\Phi + {\nabla _\mu }\Phi {\nabla ^\mu }\Phi - \frac{1}{{24}}{H_{\mu \nu \lambda }}{H^{\mu \nu \lambda }}} = 0 . $$
The first of these equations is a corrected form of the vacuum EFE, and the remaining equations represent analogous equations for other fields.
