In my view, passing to simplicial objects is all about introducing higher-dimensional structure, which gives us "maneuvering room" to do some useful things. For instance, not every $k$-algebra is a free $k$-algebra (i.e. a polynomial algebra over $k$), but every $k$-algebra is weakly homotopy equivalent to a simplicial $k$-algebra that is degreewise free. The construction is essentially a non-additive version of the construction of free resolutions in homological algebra: one way of thinking about the higher-dimensional data is to regard the $(n+1)$-simplices as specifying "relations" between the $n$-simplices.
There is a precise sense in which we should think of simplicial objects as being instructions for gluing things together. Let $\mathcal{C}$ be a locally small category with colimits for countable diagrams. The geometric realisation of a simplicial object $X_{\bullet}$ in $\mathcal{C}$ with respect to a functor $T : \mathbf{\Delta} \times \mathcal{C} \to \mathcal{C}$ is the object $\left| X \right|$ in $\mathcal{C}$ equipped with a bijection
$$\mathcal{C} (\left| X \right|, Y) \cong [\mathbf{\Delta}^\mathrm{op} \times \mathbf{\Delta}, \mathbf{Set}] (\mathbf{\Delta} (-, \bullet), \mathcal{C} (T (-, X_{\bullet}), Y))$$
that is natural in $Y$, where $\mathbf{\Delta} (-, \bullet) : \mathbf{\Delta}^\mathrm{op} \times \mathbf{\Delta}, \mathbf{Set}$ is the usual hom functor and $T (-, X_{\bullet}) : \mathbf{\Delta} \times \mathbf{\Delta}^\mathrm{op} \to \mathcal{C}$ is the functor $(n, m) \mapsto T (n, X_m)$. This turns out to amount to saying that $\left| X \right|$ fits into a certain coequaliser diagram in $\mathcal{C}$ of the form below,
$$\coprod_{\phi : [n] \to [m]} T (n, X_m) \rightrightarrows \coprod_n T (n, X_n) \to \left| X \right|$$
where the upper arrow is defined on components by $T (\phi, X_m) : T (n, X_m) \to T (m, X_m)$ and the lower arrow is defined on components by $T (n, \phi^*) : T (n, X_m) \to T (n, X_n)$.
The choice of $T : \mathbf{\Delta} \times \mathcal{C} \to \mathcal{C}$ depends on the context. For example, for $\mathcal{C} = \mathbf{Top}$, we take $T (n, Z) = \Delta^n \times Z$, where $\Delta^n$ is the standard $n$-simplex. If $\mathcal{C}$ is the category of simplicial objects in a locally small category $\mathcal{A}$ with countable colimits, then it is standard to take $T (n, Z)_m = \mathbf{\Delta} (m, n) \odot Z_m$ (i.e. $\mathbf{\Delta} (m, n)$-many copies of $Z_m$). In this situation, simplicial objects in $\mathcal{C}$ are bisimplicial objects in $\mathcal{A}$, and it turns out that for any bisimplicial object $X_{\bullet, \bullet}$ in $\mathcal{A}$, the geometric realisation is given by $\left| X \right|_n = X_{n, n}$. Thus, if $X_{\bullet, \bullet}$ is a simplicial diagram of "good" simplicial objects in $\mathcal{A}$, then $\left| X \right|_{\bullet}$ is also a "good" simplicial object in $\mathcal{A}$. That is one reason why we don't have to pass to bisimplicial objects to resolve simplicial objects.
If that's still too abstract, it might be helpful to focus on the case of simplicial objects in algebra. Let $\mathcal{A}$ be a category with a "forgetful" functor $U : \mathcal{A} \to \mathbf{Set}$ and let $\mathbf{s} \mathcal{A}$ be the category of simplicial objects in $\mathcal{A}$. Of course, we get a "forgetful" functor $U : \mathbf{s} \mathcal{A} \to \mathbf{sSet}$, so we can declare a morphism in $\mathbf{s} \mathcal{A}$ be a weak equivalence if its image in $\mathbf{sSet}$ is a weak homotopy equivalence. If $U : \mathcal{A} \to \mathbf{Set}$ has a left adjoint $F : \mathbf{Set} \to \mathcal{A}$, then $U : \mathbf{s} \mathcal{A} \to \mathbf{sSet}$ also has a left adjoint $F : \mathbf{sSet} \to \mathbf{s} \mathcal{A}$. We think of objects in the image of $F : \mathbf{Set} \to \mathcal{A}$ as being "free" objects. At this point, it is easy to show that every object $A$ in $\mathcal{A}$ can be resolved by a degreewise "free" simplicial object, i.e. there is a weak equivalence $X_{\bullet} \to A$ in $\mathbf{s} \mathcal{A}$ where $X_{\bullet}$ is a simplicial object such that each $X_n$ is a "free" object. Moreover, in favourable contexts (e.g. when $U : \mathcal{A} \to \mathbf{Set}$ factors through the forgetful functor $\mathbf{Grp} \to \mathbf{Set}$, or slightly more generally, when $U : \mathbf{s} \mathcal{A} \to \mathbf{sSet}$ factors through the full subcategory of Kan complexes), it can be shown that $F : \mathbf{sSet} \to \mathbf{s} \mathcal{A}$ preserves weak equivalences, and it then follows that every simplicial object in $\mathcal{A}$ can be resolved by a degreewise "free" simplicial object.
