Shannon Entropy to Min-Entropy
This is a consequence of the asymptotic equipartition property (AEP), which is a form of the law of large numbers. The AEP states that if a random variable has (binary) entropy $H$ and you take $n$ copies of it, then most of the data points have probability roughly $2^{-nH}$. This is only true for most data points, which is the source of the small statistical distance that you mention.

As an example, consider a $p$-biased coin. If you throw $n$ coins with bias $p$, then most likely you will get roughly $pn$ heads, each of which events has probability roughly
$$ p^{pn} (1-p)^{(1-p)n} = 2^{-nh(p)} .$$