I'm attempting to learn about string theory through a series of lectures by Freddy Cachazo. Starting 43 minutes into part 6, he begins to explain why we can associate the traceless, massless, closed string excitation with gravitation.

The Polyakov action is modified so that the metric tensor of the target space is some $G_{\mu\nu} \left(X\right)$ rather than $\eta_{\mu\nu}$, and it is shown that, in an appropriate limit, the vanishing of the world-sheet field theory's beta function (at one loop) due to conformality yields the vacuum Einstein field equations for the target space metric.

At this point, he seems to imply that the argument is finished, but I am a bit lost, as there seems to be a logical step that I am missing. Why does the emergence of the vacuum EFEs in this context tell us that this particular excitation of the string (which we found starting from the original Polyakov action) is a graviton?