This might become clearer when you know that you can obtain the category of sheaves from the category of presheaves by localizing: you invert maps that are isomorphisms locally for the topology. The localization functor can then be identified with sheafification. Note that this description of the category of sheaves makes no reference to the sheaf condition at all.
Now for simplicial sheaves, there is an obvious way to define the correct notion of a "local equivalence of simplicial presheaves": it should just be a map which, locally in the topology, is a weak equivalence of simplicial sets. So the category of simplicial sheaves, whatever it is, should be the localization (preferably in the $\infty$-categorical sense) of the category of simplicial presheaves at these maps. Hypercovers are special cases of local equivalences, and it turns out that there are enough of them: localizing at hypercovers gives you the same category of sheaves. If you localize only at Cech covers, however, you get something different.
This is why for example Cech cohomology does not always coincide with sheaf cohomology (and any site in which they disagree gives you an example where the Cech localization is different), but sheaf cohomology is always the same as hypercover cohomology.
This paper by Dugger, Hollander, and Isaksen, is a good reference for hypercovers. Theorem 6.2 is the fact that I mention above.
EDIT: I was wrong when I mentioned Cech cohomology above. Cech cohomology has nothing to do with either localization (see this relevant nForum post).