"In order for a string theory to be consistent, the worldsheet theory must be conformally invariant. The obstruction to conformal symmetry is known as the Weyl anomaly and is proportional to the central charge of the worldsheet theory. In order to preserve conformal symmetry the Weyl anomaly, and thus the central charge, must vanish." ... "In string theory, conformal symmetry on the worldsheet is a local Weyl symmetry and the anomaly must therefore cancel if the theory is to be consistent." ... "The required cancellation implies that the spacetime dimensionality must be equal to the critical dimension which is either 26 in the case of bosonic string theory or 10 in the case of superstring theory."
Individually, I have a rudimentary understanding the highlighted concepts. I've truly studied every Leonard Susskind theoretical physics lecture on youtube. Yet, I can not seem to find out anywhere just exactly how and why a Weyl anomaly links up to the concept of central charge, or how central charge is analogous to critical dimensionality, for that matter. Even Susskind never discusses it.
I think if someone could give me even a non-rigorous handle on this, I might finally feel satisfied why ten dimensions in superstring theory is so important. Note: Susskind did explain why 26 dimensions in Bosonic String Theory was critical, btw. https://youtu.be/-I7PjKyCnI0