Why Weyl invariance is important for consistent string theory?

This post is related to this link. I know there is a Weyl invariance for the Polyakov action at least in classical level. My question arises from obtaining effective action in string theory, such as section 7.3 in this lecture note

A consistent background of string theory must preserve Weyl invariance, which now requires $\beta_{\mu\nu}(G)=\beta_{\mu\nu}(B)=\beta(\Phi)=0 $

later the lecture note tried to find effective action by requiring vanishing beta function.

Also in p110 in Polchinski's string theory vol I,

We have emphasized that Weyl invariance is essential to the consistency of string theory.

Why Weyl invariance is so important for consistency of string theory?