Viewing the coordinates of spacetime as fields on string worldsheet, the strings are described by the Polyakov action which presents conformal symmetry (including others) at the claasical level.
Now this classical conformal symmetry has anomaly in quantum theory and to make the anomaly be cancelled the spacetime must have particular dimensions. My question is, why the conformal anomaly must be cancelled? Why do we care conformal symmetry so much at the quantum level ?
The Polyakov action reduces to the Nambu-Goto action when you use the gauge freedom to eliminate the $g_{ab}$ world-sheet metric. The Nambu-Goto action describes a string moving in a target space.
If you lose the Weyl gauge symmetry by an anomaly you cannot eliminate all the $g_{ab}$ anymore. Diff- will eliminate only $2$ components of $g_{ab}$. If you fix the Conformal Gauge you get:
$$ g_{ab}=e^{2\omega}\delta_{ab} $$ where $\omega(\sigma)$ is a new field in your theory (a new "direction" on the target space).
The quantum corrections that break the Weyl-symmetry will give a kinetic term to this residual degree of freedom:
$$ Z[e^{2\omega}\delta_{ab}]=Z[\delta_{ab}]\exp(\frac{c}{24\pi}\int d^2\sigma\, \delta^{ab} \partial_{a} \omega\partial_{b}\omega) $$
and this is not a Nambu-Goto theory anymore. This theory is actually the Linear Dilaton CFT and does not have the Lorentz symmetry in the $\omega$ -direction.
