One recovers Einstein's equations by considering the bosonic closed string theory with a zero B-field, a constant dilaton $\phi$ and by taking the limit $\alpha' \rightarrow 0$, $g_s \rightarrow +\infty$ such that $g_s \sqrt{\alpha'}$ is some constant $l_p$. Here $\alpha'$ is the Regge's slope i.e. the inverse of the string tension up to a normalization, $g_s$ is the string coupling constant related to the dilaton background by $g_s = e^\phi$. In this limit, the only excitation of the theory is a massless symmetric rank 2 tensor field $h_{\mu \nu}$ and the statement is that its $n$-point correlation functions coincide with the $n$-points correlations functions of the graviton computed at tree level from the Einstein-Hilbert action with the Planck length equals to $l_p$.

The original papers for this result are by Yoneya

http://ptp.oxfordjournals.org/content/51/6/1907.refs

and Scherk, Schwarz

http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?196800234

The relation between this derivation of the Einstein's equation and the one through the $\beta$-function presented in the answer of joshphysics is nicely explained in the section 10.1 of the Polyakov's book "Gauge fields and strings". The low energy effective action for the massless excitations of the string can be computed from the correlation functions of the corresponding vertex operators. These vertex operators are constructed form the weight 2 local operators of the world-sheet CFT and the spacetime correlation functions of the massless excitations can be expressed in terms of the correlation functions of these operators in the 2d CFT. But in the other hand, weight 2 local operators define the marginal deformations of the 2d CFT and the beta function with respect to one of the massless field is obtained from the variation of the theory under the corresponding deformation. To compute this variation, one can perturbatively expand the deformation of the action and one has to compute n-point correlations functions of the weight 2 local operators in the 2d CFT, i.e. the same object as for the spacetime amplitudes. One is simply saying that the deformations of the space-time background by turning on massless fields correspond to the marginal deformations of the 2d worldsheet CFT.