First of all Muro's answer is perfect. This a long comment which I hope it will be helpful, it is more about intuition.
I think Rezk's answer Group completion theorem is closely related to your question.
For any (small category) $\mathsf{C}$ and any sub category $\mathsf{W}$, Dwyer and Kan defined a simplicial localization $L_{\mathsf{W}}\mathsf{C}$ (i.e. it a simplicially enriched category which same set of objects as $\mathsf{C}$) which has the following properties:

There is an obvious functor $\mathsf{C}\rightarrow L_{\mathsf{W}}\mathsf{C}$ such that

a) $\mathsf{C}[\mathsf{W}^{-1}]\simeq \pi_{0}L_{\mathsf{W}}\mathsf{C}$

b) For any morphism $a\rightarrow b\in \mathsf{W}$, we have
$$Map_{L_{\mathsf{W}}\mathsf{C}}(b,c)\rightarrow Map_{L_{\mathsf{W}}\mathsf{C}}(a,c)$$
and
$$Map_{L_{\mathsf{W}}\mathsf{C}}(c,a)\rightarrow Map_{L_{\mathsf{W}}\mathsf{C}}(c,b)$$
are equivalences of simplicial sets.

c) The $N_{\bullet}\mathsf{C}$ is equivalent to the (coherent) Nerve $N_{\bullet}L_{\mathsf{W}}\mathsf{C}$

In the case where $\mathsf{W}=\mathsf{C}$, the simplicial category $L_{\mathsf{W}}\mathsf{C}$ is an $\infty$-goupoid.

An other categorical way to construct $L_{\mathsf{W}}\mathsf{C}$ is the following:
Take the push out in the model category $\mathbf{Cat}_{\mathbf{sSet}}$ (the category of all small simplicial categories):

$$colim[\overline{1}_{\mathsf{W}}\leftarrow 1_{\mathsf{W}}\rightarrow \mathsf{C}]$$

where $1_{\mathsf{W}}$ is described as follows, it is a disjoint union indexed by morphisms in $\mathsf{W}$ of the category of the form $\bullet\rightarrow\bullet$, and the category $\overline{1}_{\mathsf{W}}$ is described as follows, it is a disjoint union indexed by morphisms in $\mathsf{W}$ of simplicial category $I$, the simplicial category $I$ has two objects and all its mapping spaces $Map_{I}$ are contractible. Moreover, the morphism $\overline{1}_{\mathsf{W}}\leftarrow 1_{\mathsf{W}}$ is a cofibration in $\mathbf{Cat}_{\mathbf{sSet}}$. Then the simlplicial category $L_{\mathsf{W}}\mathsf{C}$ is zigzag equivalent to $colim[\overline{1}_{\mathsf{W}}\leftarrow 1_{\mathsf{W}}\rightarrow \mathsf{C}]$ as simplicial categories. Applying the (coherent Nerve) to the diagram $colim[\overline{1}_{\mathsf{W}}\leftarrow 1_{\mathsf{W}}\rightarrow \mathsf{C}]$ (which is a homotopy colimit actually, since $\mathbf{Cat}_{\mathbf{sSet}}$ is left proper) we obtain
a homotopy pushout in the category of simplicial sets (with Quillen model structure). This is not obvious since the nerve is not well behaved with respect to colimites. Then you conclude that the nerve $N_{\bullet}C\rightarrow N_{\bullet}L_{\mathsf{W}}\mathsf{C}$ is a weak equivalence of simplicial sets since
$N_{\bullet}1_{W}\rightarrow N_{\bullet}\overline{1}_{W}$ is a trivial cofibration in simplicial sets (which is left proper). Your first question (before you edited it) can be summarized to the effect of the nerve functor to the following diagram
$C\rightarrow [L_{C}C]_{0}\rightarrow L_{C}C \rightarrow \pi_{0}L_{C}C= C[C^{-1}]$
where the last morphism is the localization morphism (functor). So, only the map $N_{\bullet}C\rightarrow N_{\bullet}L_{C}C$ is an equivalence.

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