If $\mathcal{A}$ is an abelian category, then the Dold-Kan correspondence supplies an equivalence between the category of simplicial objects of $\mathcal{A}$ and the category of nonnegatively graded chain complexes in $\mathcal{A}$. One can therefore think of simplicial objects as a generalization of chain complexes to non-abelian settings.
In homological algebra, chain complexes often arise by choosing ``resolutions'' of objects $X \in \mathcal{A}$: that is, chain complexes
$$ \cdots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0$$
with homology $$H_{i}( P_{\ast} ) = \begin{cases} X & \text{ if } i = 0 \\
0 & \text{ otherwise }. \end{cases}$$
Among these, a special role is played by projective resolutions: that is, resolutions where each $P_n$ is a projective object of $\mathcal{A}$.
If $X_{\ast}$ is a simplicial space, it might be helpful to think of $X_{\ast}$
as a resolution of the geometric realization $| X_{\ast} |$. It plays the role of a ``projective resolution'' if $X_{\ast}$ is degreewise discrete: that is, if it is a simplicial set.
To answer your question I'll need to do a fairly long digression on homotopy limits and colimits. Before I delve deep into the topic let me say that there's more than one way to describe this topic, for example some people like model categories, other people might prefer triangulated categories (shudder), I'll simply explain the point of view that has been most helpful for me in the past.
The language of $\infty$-categories
I think that one of the best way to understand homotopy limits and colimits is in the settings of $\infty$-categories. In fact, in the theory $\infty$-categories then homotopy limits and colimits are just like ordinary limits and colimits and I will stop prefixing everything with the word "homotopy" from now on.
My favourite catchphrase is that $\infty$-categories are just categories together with a notion of homotopy beteween morphisms. In fact you need a bit more, basically for every two objects $X$ and $Y$ and every $n\ge0$ you need a set $\textrm{Map}(X,Y)_n$ that morally corresponds to maps $X\times \Delta^n\to Y$ (the so-called higher homotopies) satisfying a set of compatibilities that I will not detail here (a good choice is to ask that $\textrm{Map}(X,Y)_\bullet$ form a Kan complex). To keep this concrete, examples are
- Spaces, with the obvious definition of homotopy;
- Manifolds and embeddings, with the notion of homotopy given by isotopies;
- Chain complexes with the notion of chain homotopy;
- Categories, where an homotopy between two functors is a natural isomorphism (and higher homotopies just chains of $n$ composable natural isomorphisms $F_0\to F_1\to\cdots\to F_n$).
(The last example is of course the most relevant for your question).
The important part about $\infty$-categories is that the notion of homotopy should seep through in our definition of commutative diagrams (so-called coherently commutative diagrams or coherent diagrams for short). For example a commutative square
$$\require{AMScd} \begin{CD}
A @>{f}>> B\\
@V{h}VV @VV{g}V\\
C @>>{k}> D
\end{CD}$$
is the datum of four objects, four maps and a homotopy between $gf$ and $kh$.
Once you have set up all of this you can go on and define initial and terminal objects, limits, colimits and so on and so forth exactly like you did for ordinary categories (that is without homotopies). These are often called "homotopy limits" and "homotopy colimits" to distinguish from the limits and colimits computed without paying attention to the homotopies.
Ok, but what's the deal with the Čech complex anyway?
Let's give an example. Suppose that $C,D$ are two categories and $F,G:C\to D$ two functors. If we ignored the homotopies the equalizer would simply be the objects $c\in C$ such that $Fc=Gc$. But we all know that this is a stupid notion. Naturally equivalent functors will give different answers and we do not want to distinguish between naturally equivalent functors. So we use the notion of equalizer in the $\infty$-category of categories.
As you can see immediately from studying the coherent diagrams of the form $*\to C\rightrightarrows D$ the objects of the equalizer in this brave new setting are objects $c\in C$ together with an isomorphism $\alpha:Fc\cong Gc$. Now that's a much better behaved notion!
Similarly, you can see that if $F:\Delta^{op}\to C$ is a functor to an ordinary category we get that
$\lim_{\Delta^{op}} F = \textrm{eq}(F([0])\rightrightarrows F([1]))$
so that $F(X)\to \lim F(\check U_\bullet)$ being an equivalence is just the ordinary sheaf condition. But what happens if we go one categorical level up? Now let $F:\Delta^{op}\to \mathrm{Cat}$ be a functor to categories. Then you can see that an object of $\lim_{\Delta^{op}}F$ is an object $x\in F([0])$ together with an isomorphism $\alpha:d_0^*x\cong d_1^*x$ in $F([1])$ such that $d_0^*\alpha\circ d_2^*\alpha = d_1^*\alpha$. But this is exactly the notion of descent datum.
So the notion of sheaf in this brave new context is just the clasical notion of stack
You might have noticed that we are using just a small portion of the diagram. This is because our categories do not have many "interesting" homotopies. Sets have no homotopies at all and categories have only interesting 1-homotopies, higher homotopies are basically composable sequences of 1-homotopies. In general, the higher the order of the interesting homotopies, the more pieces of $\Delta$ we have to use. (precisely for an $n$-category you just need to go to $[n+1]$. The reason for this is basically Quillen's theorem A, as noted by Dylan Wilson in the comments). For spaces and chain complexes you need to consider the whole of $\Delta$.
Conclusions and references
Ugh that's a long answer. I hope I managed to give at lesat an inking of what's going on without getting bogged by the technical details. If technical details are what you want however the standard references for $\infty$-categories are Luries's Higher Topos Theory and Higher Algebra. They are not an easy read, by anyone's standards.
The model of $\infty$-categories I've used is called fibrant simplicial categories. I think it is an excellent model to develop some intuition, but it is actually quite bad to work with. Most of the people in the industry use quasicategories, which are quite pleasant to work with but I didn't have the time to properly introduce them here.
Appendix: why can we truncate?
Let $E$ be an $n$-category, that is an $\infty$-category such that the mapping spaces $\mathrm{Map}(x,y)$ are $n$-truncated for every $x,y\in E$. Let $j:C\to D$ be a functor such that for every $d\in D$ the geometric realization $|C\times_D D_{d/}|$ is $n$-connected. Then for every functor $F:D\to E$ $\lim_D F$ exists if and only if $\lim_C Fj$ exists and they coincide.
Lemma 1: Let $K$ be a simplicial set such that the geometric realization $|K|$ is $n$-connected. Then for every $e\in E$ the limit of the constant functor $K\to E$ at $e$ exists and coincides with $e$.
Proof: $\mathrm{Map}(e',\lim_K e) \cong \lim_K \mathrm{Map}(e',e) = \mathrm{Map}(K,\mathrm{Map}(e',e))=\mathrm{Map}(e',e)$.
Lemma 2: Let $p:\tilde D\to D$ be a cartesian fibration such that for every $d\in D$ the geometric realization $|\tilde D_d|$ is $n$-connected. Then for every functor $F:D\to E$ the limit $\lim_D F$ exists if and only if the limit $\lim_{\tilde D} Fp$ exists and they coincide.
Proof: By an easy cofinality argument the right Kan extension along a cartesian fibration is obtained by computing the limits fiberwise. Then the thesis follows from Lemma 1.
Proof of the main result: Let $\tilde D\to D$ the cartesian fibration classified by the functor $D^{op}\to \mathrm{Cat}$ given by $d\mapsto C\times_D D_{d/}$. We have a canonical functor $C\to \tilde D$ sending $c$ to $(c,jc=jc)$. A standard cofinality argument implies that the functor $C\to \tilde D$ is coinitial. Then the thesis follows from the previous lemma.
Best Answer
EDIT Corrected a couple of inaccuracies and mistakes, added some references.
For the sake of clarity, let me work with non-negatively graded cochain complexes, and analyze the case of a left exact covariant functor, to prove that its right derived functor is a universal (covariant) cohomological $\delta$-functor.
Let $\mathcal A$ and $\mathcal B$ be two abelian categories, and let $F : \mathcal A \to \mathcal B$ be a left exact functor. Let me denote by $\gamma_\mathcal{A}$ the localization functor $\mathrm{Ch}_+(\mathcal{A}) \to \mathrm{D}_+(\mathcal{A})$. Let $\tilde S_\mathcal{A}$ denote the full subcategory of $\mathrm{Ch}_+(\mathcal A)$ consisting of discrete complexes (i.e. of complexes whose differentials are all zeroes). Let $S_\mathcal{A}$ denote its essential image in the derived category.
First of all, observe that a $\delta$-functor is a particular case of what MacLane calls "connected sequences" (connected sequence is a $\delta$-functor iff the long sequence it induces is exact). Now, defining a connected sequence is equivalent to define a functor $S_\mathcal{A} \to \mathcal{B}$ (see Proposition XII.8.1 and Proposition XII.8.2 in [1]). Part of the equivalence works as follows: given any $T : S_\mathcal{A} \to \mathcal{B}$, define $T^n(A) := T(A[n])$, and check that, given any exact sequence $0\to A\to B\to C\to 0$, the image of the class corresponding to it in $$S_\mathcal{A}(C[n],A[n+1]) \simeq \mathrm{D}_+(\mathcal A) (C[n], A[n+1]) \simeq \mathrm{Ext}^1_{\mathcal A}(C,A)$$ under $T$ gives you a morphism $\delta^n : T^n C \to T^{n+1} A$ giving $\{T^n\}$ the structure of a connected sequence. You can find the details of this construction in [1].
We have that $$\mathbb{R}F := \mathrm{Lan}_{\gamma_\mathcal{A}(-[0])} (\gamma_\mathcal{B} F(-)[0]).$$ Now, we can postcompose the bottom corners of the square defining $\mathbb{R}F$, along the functors $$\gamma_{\mathcal A} \circ \bigoplus_{n\geq 0} H^n : \mathrm{D}_+(\mathcal{A}) \to S_\mathcal{A}$$ and $$H^0 : \mathrm{D}_+(\mathcal{B}) \to \mathcal{B}$$ as follows
$$ \require{AMScd} \begin{CD} \mathcal{A} @>{F}>> \mathcal{B}\\ @VVV @VVV \\ \mathrm{D}_+(\mathcal{A}) @>{\mathbb{R}F}>> \mathrm{D}_+(\mathcal{B})\\ @VVV @VVV \\ S_\mathcal{A}@. \mathcal{B} \end{CD} $$
and further Kan extend, obtaining the connected sequence $RF : S_\mathcal{A} \to \mathcal{B}$ as $$\mathrm{Lan}_{\gamma_\mathcal{A} \circ \oplus_n H^n \circ (-)[0]}(H^0 \circ (-)[0] \circ F) \simeq \\ \simeq \mathrm{Lan}_{\gamma_\mathcal{A} (-[0])}(F).$$ Now, given any other connected sequence (in particular, any $\delta$-functor) $T : S_\mathcal{A} \to \mathcal{B}$, by definition of left Kan extension we have $$\mathrm{Nat}(RF,T) \simeq \\ \simeq \mathrm{Nat}(F, T \circ \gamma_{\mathrm{A}} (-[0])) \simeq \\ \simeq \mathrm{Nat}(F,T^0)$$ showing that the "universality" of $RF$ works not only for $\delta$-functors, but in general for connected sequences.
The fact that for $F$ left exact and $\mathcal A$ with enough injectives one has that $RF$ is not only a connected sequence, but really a $\delta$-functor, follows from XII.8.3, 4 and 5 in [1].
[1] MacLane - Homology