The Stacks project

Proof. A scheme $X$ over $K$ is étale over $K$ if and only if $X \cong \coprod _{i\in I} \mathop{\mathrm{Spec}}(K_ i)$ with each $K_ i$ a finite separable extension of $K$ (Morphisms, Lemma 29.36.7). The functor of the lemma associates to $X$ the $G$-set

with its natural left $G$-action. Each element has an open stabilizer by definition of the topology on $G$. Conversely, any $G$-set $S$ is a disjoint union of its orbits. Say $S = \coprod S_ i$. Pick $s_ i \in S_ i$ and denote $G_ i \subset G$ its open stabilizer. By Galois theory (Fields, Theorem 9.22.4) the fields $(K^{sep})^{G_ i}$ are finite separable field extensions of $K$, and hence the scheme

is étale over $K$. This gives an inverse to the functor of the lemma. Some details omitted. $\square$