Intuition for Krasner's Lemma
Be careful - your statement of Krasner's Lemma contains a grammatical ambiguity. The statement should be:
Let $K$ be a local field complete with respect to a nontrivial nonarchimedean valuation, and let $K^\mathrm{al}$ be an algebraic closure of $K$. Let $\alpha, \beta \in K^\mathrm{al}$, where $\alpha$ is separable over $K(\beta)$. If $|\beta - \alpha| < |\sigma\alpha - \alpha|$ as we run through all conjugates $\sigma \alpha \neq \alpha$ over $K$, then $K(\alpha) \subseteq K(\beta)$.
(I don't think discreteness of the valuation is used anywhere in the proof.)
Picture the $\sigma \alpha$ as points in $K^\mathrm{al}$, each with its own neighborhood. Krasner says that, for all $\beta$ in a small enough neighborhood of $\alpha$, the only conjugate of $\alpha$ over the base field $K(\beta)$ is $\alpha$ itself, so that every automorphism in $\mathrm{Gal}(K^\mathrm{al}/K)$ that fixes $\beta$ must also fix $\alpha$.