AR Modeling: Why residual is white noise?
it's a hand-wavy argument, but the idea is to derive the coefficients $\alpha_i$ so that the norm of the error $\Big\||\epsilon[n]|^2\Big\|$ is minimized.  we assume that $x[n]$ is a stationary "random" process, we know all of the previous samples $x[n-i]$ and we want to make a good guess at $x[n]$ with a linear combination of the previous samples.  we can only make some guess of the next $x[n]$ if it is not a white random process.  if it is white, knowing the previous samples is not going to help us at all.  if it is not white, then the autocorrelation of $x[n]$ is not zero for non-zero lags and from that we can compute $\alpha_i$.

now if $\epsilon[n]$ was not white, we could do further LPC to make a good guess at $\epsilon[n]$ and make a better guess for $x[n]$, but if the coefficients $\epsilon[n]$ are optimal, then we can't make a better guess for $x[n]$.  so then the error of our guess for $x[n]$ and what we really get has to be white.