As you said, it is a generalization of Cech theory. The standard example to understand first is a divisor $D=\bigcup D_i$ with simple normal crossings. A resolution of singularities is obtained by simply taking a disjoint union of components $X_0= \coprod D_i$. Let $\pi_0:X_0\to D$ be the obvious map. The cohomology of $X_0$ with your favourite coefficients will not (usually) be the same as $D$. So you want to correct this by adding in "higher simplicies". Let $X_1$ be the disjoint union $\coprod D_i\cap D_j$. This has a pair of "face" maps to $X_1\to X_0$. We can continue this process to get a (strict) simplicial object
$$ \ldots X_1\rightrightarrows X_0\to D$$
with an augmentation to $D$. Given a sheaf $F$ on $D$, we can pull it back to a get a collection of sheaves $F_i$ on $X_i$ with various structure maps. We can define the cohomolgy
$H^i(X_\bullet, F_\bullet)$ by taking (for example) by compatible injective resolutions, applying $\Gamma$, and taking cohomology of the total complex of the resulting double complex. Now the point is that the machinery of descent tells you that
$$H^i(D, F)\cong H^i(X_\bullet, F_\bullet)$$
or if you prefer, there is a spectral sequence relating $H^*(X_p, F_p)$ and $H^*(D,F)$.
This reduces down to a Mayer-Vietoris type sequence
$$\ldots H^i(D,F)\to H^i(D_1, F)\oplus H^i(D_2,F)\to H^i(D_1\cap D_2, F)\ldots$$
when $D$ has two components. Why is this good? Because for certain things, e.g.constructing mixed Hodge structures, it's better to replace the singular space by a bunch of smooth spaces.

As per request, I'm expanding my comment, although this may be a bit too concise. If $X_\bullet\to X$ and $Y_\bullet\to X$ are two
simplicial resolutions, take the fibre product to get simplicial scheme
$Z_n =\coprod_{i+j=n}X_i\times_X Y_j$ dominating both. However, it will be singular. You can build simplicial resolution $\tilde Z_\bullet$ of this inductively.
First resolve singularites of $Z_0$ to get $\tilde Z_0$. For the next step, choose a resolution of $Z_1$ which maps to $\tilde Z_0\times_{X}\tilde Z_0$ etc.