Infinity local systems
One way to discuss $\infty$-local systems over a space $X$ is in terms of the fundamental $\infty$-groupoid of $X$. To motivate this, recall that for classical local systems, one has the equivalence of categories $$LocSys(X) \simeq Rep(\pi_{1}(X)) $$ between the local systems on $X$ and representations of the fundamental group. It's easily verified that representations of the fundamental group are equivalent to functors out of the fundamental groupoid of $X$, so it's more natural (in the sense of not making any choices) to write $$LocSys(X) \simeq Fun(\Pi_{1}(X), Vec_{k}) $$
The idea behind $\infty$-local systems is then to replace vector spaces with chain complexes, which we think of as a homotopical/derived version of vector spaces. The problem with just doing this naively though (say, by taking $Fun(\Pi_{1}(X), Ch_{k})$ where $Ch_{k}$ is the category of chain complexes over $k$), is that the fundamental groupoid only encodes information about the 1-truncated homotopy type of $X$. If we're mapping into a category that carries higher homotopical data, there's no way we're going to get a satisfactory answer unless we encode the entire homotopy type of $X$ in the domain.
So instead, we replace $\Pi_{1}(X)$ by $\Pi_{\infty}(X)$ - which is probably more familiarly written as the singular simplicial set associated to $X$, denoted $Sing(X)$. Then, we can take as our definition of infinity local systems $$LocSys^{\infty}(X) := Fun(\Pi_{\infty}(X), Ch_{k})$$ where we view $Ch_{k}$ as an $\infty$-category. I'm not entirely sure how this definition fits in with that of the paper you cited, but it's certainly a common way to think of infinity local systems.
Some places to read:
- Appendix A of Higher Algebra. This is definitely overkill for your question, as it mostly pertains to $\infty$-constructable sheaves, but it's a great place to learn some sheaf theory from the above perspective.
- Kerodon, Section 2.5. This isn't directly about infinity local systems, but gives a lot of insight into the interplay between dg-categories and $\infty$-categories. (In particular, this process of 'taking chains' of a simplicial set mentioned in the comments by Dylan is explained here in detail).
As for your second question, the comments give a pretty concrete and conceptual answer as to how to perform this computation, but there's another heuristic that explains why you would expect this to be true without computing anything. The key point is that the dg-category $LocSys^{\infty}(X)$ is a differential graded enhancement of the derived category of local systems on X. This means that when we pass to the homotopy category, we recover the derived category of local systems.
Now in the (underived, 1-categorical) category of local systems, $hom(k_{X}, \mathcal{F})$ is equivalent to just taking global sections of $\mathcal{F}$. So, when we pass to the derived category, $hom(k_{X}, k_{X})$ is just taking derived global sections of the constant sheaf - i.e. singular cohomology of $X$ with coefficients in $k$. Thus, in the dg-category $LocSys^{\infty}(X)$, the mapping complex $hom(k_{X}, k_{X})$ is some chain complex whose $i^{th}$ cohomology is isomorphic to $H^{i}(X ; k)$. So it shouldn't be surprising at all that this is $C^{\star}(X; k)$.
Since you tagged this with "symplectic geometry", I'm going to give an answer from a symplectic geometry perspective, which may not be what you're looking for, but (as a symplectic person) I find it is a helpful point of view. This will use the language of $A_\infty$-categories as well as dg-categories.
Given a manifold $X$, take its cotangent bundle $T^*X$. This is a non-compact symplectic manifold. You can consider its Fukaya category of exact embedded Lagrangian submanifolds which agree with the symplectisation of a Legendrian at infinity. The Floer cochain groups (morphisms) between $L_1,L_2$ are free k-vector spaces on intersection points between $L_1$ and $\phi(L_2)$, where $\phi$ is the time 1 map of a suitable Hamiltonian. You have to specify a suitable class of Hamiltonians, and I want to use Hamiltonians which are "quadratic at infinity", in other words they look something like kinetic energy with respect to some Riemannian metric. Since kinetic energy as a Hamiltonian generates geodesic flow, the Floer cochains are something like (k-linear combinations of) geodesics connecting the Lagrangians.
For example, if $L_1$ and $L_2$ are cotangent fibres at $x_1$ and $x_2$ then the Floer complex is something like the free k-vector space on the set of geodesics from $x_1$ to $x_2$. It is then a theorem (of Abbondandolo and Schwartz at the level of homology beefed up to the $A_\infty$ level by Abouzaid) that the Floer complex between two cotangent fibres is quasiisomorphic to chains on the space of paths between $x_1$ and $x_2$ (and to $C_{-*}(\Omega X)$ as an $A_\infty$-algebra when $x_1=x_2$ and concatenation makes sense).
Abouzaid also showed that a cotangent fibres generates this Fukaya category, so you get a fully faithful Yoneda functor from the Fukaya category to the dg-category of $A_\infty$-bimodules over chains on the based loop space. In other words, if you want to compute the Floer complex between two Lagrangians $L_1$ and $L_2$ and you have a cotangent fibre $F$, you can take the $CF(F,F)$-bimodules $CF(F,L_n)\otimes CF(L_n,F)$, $n=1,2$, and take homs between them in the category of $A_\infty$ $CF(F,F)$-bimodules. Since $CF(F,F)\cong C_{-*}(\Omega X)$, this is quasiequivalent to the category of infinity local systems. So this category of infinity local systems is the Fukaya category of $T^*X$.
Now how do you see that $hom(k,k)=C^*(X)$? Well, there is a Lagrangian in $T^*X$ whose Floer complex is $C^*(X)$, namely the zero section. For example, a small Hamiltonian deformation of the zero section is a graph of an exact 1-form $df$, and the intersection points between this and the zero section happen at critical points of $f$; in fact Floer showed that in this case the Floer complex is the Morse complex for a suitable choice of almost complex structures.
What is the Yoneda bimodule corresponding to the zero section? Well the zero section intersects our cotangent fibre at a single point, so $CF$ is just k (our field, considered as a trivial $A_\infty$-module over $C_{-*}(\Omega X)$). Its self homs in the category of $C_{-*}(\Omega X)$-bimodules should therefore compute $C^*(X)$.
The relevant papers of Abouzaid are:
https://arxiv.org/abs/0907.5606
https://arxiv.org/abs/1003.4449