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A sketch of Artin's construction (added Aug 6, 2017).

I found the following short explanation of Artin's construction of elementary neighborhoods on my hard drive. It is enough to prove:

Lemma. Let $X$ be a smooth variety over an infinite field $k$, let $x\in X(k)$ be a point. Then there exists a Zariski neighborhood $U$ of $x$ and an elementary fibration $\pi:U\to S$.

Proof (sketch). We can assume $X$ is affine, embed $X$ into a normal projective $\bar X\subseteq \mathbf{P}^N$, let $Y = \bar X - X$ be the complement. By a Bertini-type argument, after passing to a higher Veronese embedding, there exists a linear subspace $L$ of codimension $\dim X - 1$ containing $x$, avoiding the singularities of $\bar X$ and $Y$ and transverse to $\bar X$ and $Y$. We can furthermore find a subspace $C\subseteq L$ of codimension 1, not containing $x$, disjoint from $Y$ and meeting $X\cap L$ transversely. The projection $\mathbf{P}^N\setminus C\to \mathbf{P}^{\dim X - 1}$ along $C$ is a rational map which extends to the blow-up of $\mathbf{P}^N$ along $C$. Taking the proper transform $\bar X'$ of $\bar X$ in this blow-up, we get a proper surjective map $\bar f: \bar X'\to \mathbf{P}^{\dim X - 1}$ whose fiber at $\bar f(x)$ is the smooth curve $L\cap \bar X$. Then $\bar f$ defines an elementary fibration as needed.

In fact, this application is sort of a red herring, as later Artin proved a much more general result (with $X$ finite type but not necessarily smooth and $\mathscr{F}$ an arbitrary constructible etale sheaf) using reduction to curves. Still, Artin's result has had some influence on algebraic geometry, for example it showed up in Faltings' work on $p$-adic Hodge theory (cf. Olsson's paper!).

In good cases ($X$ a CW complex or locally noetherian), this is equivalent to the vanishing of $\pi_q(X)$ for $q>1$ (in the etale setting, these are the etale homotopy groups of Artin and Mazur).

Definition. A group $G$ is a good group if for every finite $G$-module $M$, the natural maps $$ H^*(\hat G, M)\longrightarrow H^*(G, M) $$ are isomorphisms (here $\hat G$ denotes the profinite completion of $G$.

(I will finish later).

In good cases ($X$ a CW complex or locally noetherian), this is equivalent to the vanishing of $\pi_q(X)$ for $q>1$ (in the etale setting, these are the etale homotopy groups of Artin and Mazur). Examples of $K(\pi, 1)$'s are smooth curves not isomorphic to $\mathbb{P}^1$ or spectra of fields.

Definition. A group $G$ is a good group if for every finite $G$-module $M$, the natural maps $$ H^*(\hat G, M)\longrightarrow H^*(G, M) $$ are isomorphisms (here $\hat G$ denotes the profinite completion of $G$).

Finite groups, free groups, and extensions of good groups are good - this will be important in a second. A nice example of a group which is not good is ${\rm Sp}(2g, \mathbb{Z})$ for $g>2$ (I don't remember the exact bound). This implies that the moduli space of principally polarized complex abelian varieties with suitable level structure is a $K(\pi, 1)$ in the topological sense but not for the etale topology.

To understand why (2) is true, it is enough to know the following:

Lemma. Let $f:X\to S$ be an elementary fibration over a regular noetherian $\mathbb{Q}$-scheme $S$. Then

(a) If $S$ is a $K(\pi, 1)$ then so is $X$,

(b) If $S$ is of finite type over $\mathbb{C}$ and $S(\mathbb{C})$ is a $K(\pi, 1)$, then $X(\mathbb{C})$ is a $K(\pi, 1)$,

(c) If $S$ is of finite type over $\mathbb{C}$, then $\pi_1(X(\mathbb{C}))$ is an extension of $\pi_1(S(\mathbb{C}))$ by a free group. In particular, if $\pi_1(S(\mathbb{C}))$ is good, so is $\pi_1(X(\mathbb{C}))$.

Assertions (b) and (c) follow easily from the (not entirely obvious but intuitively clear) fact that $f:X(\mathbb{C})\to S(\mathbb{C})$ is a locally trivial fibration whose fiber is an open Riemann surface, whose homotopy type is that of a wedge of circles. Assertion (a) can be proved directly using Abhyankar's lemma (this is where we need characteristic zero, otherwise the proof fails due to wild ramification problems) and the Leray spectral sequence, cf. Lemma 5.5 in Olsson's article. Alternatively, assertion (a) can be deduced from (b) and (c) if $S$ is of finite type over $\mathbb{C}$ using the comparison theorems between singular and etale cohomology, but that would be a wrong thing to do since Artin's original motivation was to prove that comparison!

Which brings us to why Artin proved this theorem in the first place.

Application. Let $X$ be smooth and of finite type over $\mathbb{C}$ and let $\mathscr{F}$ be an locally constant constructible sheaf on $X$. Then the natural maps $$ H^*(X, \mathscr{F})\longrightarrow H^*(X(\mathbb{C}), \mathscr{F}) $$ are isomorphisms.

If $X$ is an Artin neighborhood, then this follows from the fact that the above maps can be factored as (sorry, I forgot how to draw a commutative square on MO): $$ H^*(X, \mathscr{F}) \cong H^*(\pi_1(X, x), \mathscr{F}_x) \cong H^*(\pi_1(X(\mathbb{C}), x), \mathscr{F}_x) \cong H^*(X(\mathbb{C}), \mathscr{F}). $$ Here the first isomorphism comes from the fact that $X$ is a $K(\pi, 1)$ as a scheme, the second from the fact that $\pi_1(X(\mathbb{C}),x)$ is a good group (combined with the theorem from SGA1 which says that $\pi_1(X, x)$ is the profinite completion of $\pi_1(X(\mathbb{C}), x)$), and the last one from the fact that $X(\mathbb{C})$ is a $K(\pi, 1)$ space. If $X$ is not necessarily an Artin neighborhood, it can be covered by such, and their pairwise intersections can be covered by such, etc., and on both sides we can write a spectral sequence of a hypercovering.