Why are variables in a sample correlated but PC scores obtained from these variables are not? Consider a random sample of observations X = [X1, . . . , Xn]. Why would we expect that, typically, the variables of the sample X are correlated, but the principal component scores obtained from these variables are not?

We don’t necessarily expect the original features to be uncorrelated, such as in a designed experiment where they are set up specifically to lack any observed correlation. For random or observational features, however, we expect at least a little sample correlation largely because there are a lot more options for being at least slightly correlated (all numbers $-1$ to $+1$ except $0$) than for being totally uncorrelated (exactly $0$).
For the principal components, however, they are required to be uncorrelated by construction. The process of PCA specifies a diagonal covariance matrix, meaning zero covariance between features, thus zero correlation.
