Actually, for simplicial sheaves, and to be more accurate infinity sheaves of infinity groupoids, you do not need hypercovers. Your "naive" idea about multiple intersections (actually fibered products) is correct. If you instead use hypercovers, you get the notion of a *hyper*sheaf. Both infinity sheaves and hypersheaves form an infinity topos, and the infinty topos hypersheaves is a left exact localization of the infinity topos of infinity sheaves. They COULD be the same, but this depends on your Grothendieck site. They are the same if and only if the infinity topos of infinity sheaves satisfies Whitehead's theorem internally with respect to internal homotopy sheaves; for any infinity sheaf $X$ over $\mathcal{C}$, one has sheaves $\pi_n(X) \in Sh_{\infty}\left(C\right)/X$ (it actually is a discrete object thereof) analogous to homotopy groups of spaces. The infinity topos of infinity sheaves is hypercomplete if and only if any morphism $g:X \to Y$ which induces an isomorphism on $\pi_n$ for all $n$
must be an equivalence. If any arbitrary infinity topos is not hypercomplete, we may hypercomplete it by localizing with respect to those morphisms which induces isomorphisms on all homotopy sheaves, and this is again an infinity topos, called its hypercompletion. So the more precise statement is that for an infinity topos of infinity sheaves over a site $\mathcal{C}$, its hypercompletion can be identified with the infinity topos of hypersheaves.

One should note that an infinity topos is defined to be a left exact (accessible) localization of an infinity category of infinity presheaves on some infinity category, and that this DOESN'T imply that you are the infinity topos of infinity sheaves on some site, in the infinite case; the infinity topos of hypersheaves over some site need not be equivalent to ordinary infinity sheaves over any site.

Now to answer the question in the comments below:

Why do we need the full simplicial Cech diagram for infinity sheaves?

Well, let go back to ordinary sheaves of sets first. A Grothendieck topology should actually be defined in terms of covering *sieves* not in terms of covering families. What most people call a Grothendieck topology is really a basis for a Grothendieck topology. A sieve on an object $C$ of $\mathcal{C}$ is a subobject $S$ of the representable presheaf $y\left(C\right)$ in the category $Set^{\mathcal{C}^{op}}$. If we are given a basis for a Grothendieck topology, and a cover $\mathcal{U}:=\left(f:U_\alpha \to C\right)$ of $C,$ then we can consider the sieve $S_\mathcal{U}$ to be the subobject which assigns an object $D$ the morphisms $D \to C$ which factor through some $f_\alpha$. A sieve $R$ is a called a *covering sieve* if there is some cover $V$ such that $S_V \subset R$. Now a presheaf $F$ is said to be a *sheaf* if for any covering sieve $R \subset y(C),$ the induced map $$F(C) \cong Hom(y(C),F) \to Hom(R,F)$$ is an isomorphism. One can check that this is equivalent to requiring this condition on sieve of the form $S_\mathcal{U}$ for some cover. But, $S_\mathcal{U}$ can be described as the colimit of the diagram $$\coprod\limits_{\alpha,\beta}{y\left(U_\alpha \times_C U_\beta\right)}\rightrightarrows \coprod\limits_{\alpha}{y\left(U_\alpha\right)}$$ in presheaves (you can check this easily by computing this "object-wise"). But this means that $$Hom(S_\mathcal{U},F) \cong \varprojlim \left(\prod \limits_{\alpha}{Hom(y\left(U_\alpha\right),F)} \rightrightarrows \prod\limits_{\alpha,\beta}{Hom(y\left(U_\alpha \times_C U_\beta\right),F)}\right).$$ After Yoneda, this recovers the ordinary definition of sheaf with covers.

But now one can simply say that an infinity sheaf is an infinity presheaf $F$ of infinity groupoids such that for any covering sieve $R \subset y(C),$ the induced map $$F(C) \simeq Hom(y(C),F) \to Hom(R,F)$$ is an *equivalence of infinity groupoids* (i.e. a weak homotopy equivalence of simplicial sets). Again, it suffices to check this condition for sieves of the form $S_\mathcal{U}$. Now consider the infinity colimit (i.e. homotopy colimit) of the Cech diagram coming from $\mathcal{U}$. Call it $S'$. We can compute its 0-truncation $S'_0$, and since this functor is left-adjoint to the inclusion of ordinary presheaves, the unit of the adjunction furnishes us with a map $l:S' \to S'_0$. But since it is a left-adjoint, it preserves colimits, so it is equal to the colimit of the same diagram in ordinary presheaves. But, the "2-stage" diagram is cofinal in this case, so we recover $S_\mathcal{U}$. But, the homotopy fibers of $l:S' \to S'_0=S_\mathcal{U}$ are discrete, so this means $l$ is both 0-connected and 0-truncated, hence an equivalence. But this means that $S_\mathcal{U}$ is the homotopy colimit of the whole simplicial diagram, when we consider it as an object in infinity presheaves. This in terms tells us that $Hom(S_\mathcal{U},F)$ is a homotopy *limit* of a cosimplicial diagram, and envoking the infinity Yoneda lemma, we get that this diagram is equivalent to the one involving applying $F$ to iterative fibered products.