UPDATE: I have written a more complete answer here: How do the Einstein's equations come out of string theory?
If you take the Polyakov (gravitons are bosons) Action:
SP=−T2∫√±hhαβ∂αXμ∂βXνgμν d2ξ
And take the gravitational terms of a somewhat "effective" spacetime action, you get
SG=λ∫(R+ℓ2sRμνρσRμνρσ) dDx
Where we neglected terms of order ℓ4s and greater. Since ℓs, the string length, is very small, this is approximately, (if we let ℓs→0)
SEH=λ∫R dDx
The generalised (n-dimensional) EH Action. This is what is meant by string theory going to GR at the classical limit, the Polyakov Action simply goes down to the EH Action.
Edit: Also see JoshPhysics's answer here: In what limit does string theory reproduce general relativity?. The method I stated here is possible in principle, but is much more commplicated than the JoshPhysics's answer there. In his answer, he simply uses the Beta functional, βGμν=ℓ2sRμν+ℓ4sRμνRμνρσRμνρσ+...
Then, setting the LHS to 0 to preserve conformal invariance:
Rμν+ℓ2sRμνRμνρσRμνρσ+...=0
For weak gravity, all terms except the first vanish, so that
Rμν=0