 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 ${\alpha }^{\prime }$ 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