Here is a deeply ahistorical approach. Let's begin with the following topological analogue of a Galois extension with Galois group $G$, namely a Galois cover $Y \to X$ of spaces with Galois group $G$. Let's consider what the cohomology of these things looks like (as a topological analogue of taking etale cohomology). The cohomology of $Y$ over $\mathbb{C}$ decomposes as
$$H^{\bullet}(Y, \mathbb{C}) \cong H^{\bullet}(X, \mathbb{C}[G]) \cong \bigoplus_V H^{\bullet}(X, V) \otimes V^{\ast}$$
which means the following. First, $Y$ determines a $G$-local system on $X$, and so for any representation of $G$ we get an associated local system of vector spaces. $\mathbb{C}[G]$ denotes the local system on $X$ associated to the regular representation of $G$, which then decomposes as $\bigoplus_V V \otimes V^{\ast}$ where $V$ runs over irreducible representations of $G$. Hence cohomology with local coefficients in $\mathbb{C}[G]$ decomposes as a direct sum as above, where $H^{\bullet}(X, V)$ denotes cohomology with coefficients in the local system on $X$ associated to $V$. $G$ acts on $Y$ by covering transformations, and the corresponding action on the RHS is on the factors $\otimes V^{\ast}$. I may be off by a dualization here.
I claim this is a precise topological analogue of the factorization of the Dedekind zeta function of $L$ as a product of Artin L-functions. The idea is to think of the Dedekind zeta function as something determined by the etale cohomology of $\text{Spec } \mathcal{O}_L$, but multiplicatively, so that direct sums get sent to products, and then to hope / expect that the etale cohomology has a decomposition as above.
The construction which takes as input the etale cohomology of $\text{Spec } \mathcal{O}_L$ and returns as output the Dedekind zeta function has something to do with taking traces of Frobenius on the total symmetric power
$$\text{Sym}^{\bullet}(H^{\bullet}(\text{Spec } \mathcal{O}_L))$$
of the etale cohomology. At this point let me record the following useful observation from linear algebra.
Observation: Let $V$ be a finite-dimensional vector space on which a linear operator $A$ acts. Let $\text{Sym}^k(A)$ denote the induced action of $A$ on $\text{Sym}^k(V)$. Then
$$\sum t^k \text{ tr Sym}^k(A) = \frac{1}{\det (1 - At)}.$$
So, taking for granted that the etale cohomology at least heuristically decomposes into pieces $H^{\bullet}(\text{Spec } \mathbb{Z}, V)$, the corresponding thing to do to each piece is to take the trace of Frobenius on the total symmetric power of etale cohomology with local coefficients
$$\text{Sym}^{\bullet}(H^{\bullet}(\text{Spec } \mathbb{Z}, V))$$
and this is why one shouldn't be surprised to see the characteristic polynomial of Frobenius acting on $V$ in the local factors. (Here the topological analogue of the action of Frobenius only being well-defined up to conjugation is the holonomy around a loop only being well-defined up to conjugation.)