= Solution
The worldsheet metric is a gauge variable, so quantum consistency requires the <Weyl transformation> to remain non-anomalous. The background fields are couplings of a two-dimensional quantum field theory, and their <sigma-model beta functions> multiply the trace of the worldsheet stress tensor. Requiring every beta function to vanish gives, at the first nontrivial order in the derivative or $\alpha'$ expansion, precisely the stated metric, B-field and dilaton equations. They are also the Euler-Lagrange equations of the leading spacetime string-frame effective action
$$
S_{\rm eff}\propto\int d^{26}x\sqrt{-G}\,e^{-2\Phi}
\left[R+4(\nabla\Phi)^2-\frac1{12}H^2+O(\alpha')\right].
$$
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