I apologize that this answer is well-known to any expert on Langlands, but I believe it responds to what the OP was asking:
Let $K$ be a number field and $S_K$ its set of places. The kind of Galois representations considered in the Langlands program have coefficients in $\overline{\mathbb{Q}_{\ell}}$ for a fixed prime $\ell$ and are unramified outside a finite set $S \subseteq S_K$, i.e. representations of $G_{K,S}$. For a finite place $\mathfrak{p} \in S_K \setminus S$, there is a conjugacy class of maps $G_{K_{\mathfrak{p}}} \to G_{K,S}$ factoring through the quotient $G_{K_{\mathfrak{p}}} \twoheadrightarrow \widehat{\mathbb{Z}}$ generated by $\mathrm{Frob}_{\mathfrak{p}}$. Thus for a representation $\rho \colon G_{K,S} \to \operatorname{GL}_n(\overline{\mathbb{Q}_{\ell}})$, we get a semisimple conjugacy class $\rho(\mathrm{Frob}_{\mathfrak{p}})^\text{ss}$ in $\operatorname{GL}_n(\overline{\mathbb{Q}_{\ell}})$. The Chebotarev Density Theorem tells us that this collection $\{\rho(\mathrm{Frob}_{\mathfrak{p}})^\text{ss}\}_{\mathfrak{p} \in S_K \setminus S}$ of conjugacy classes of $\operatorname{GL}_n(\overline{\mathbb{Q}_{\ell}})$ indexed by almost all places $\mathfrak{p}$ of $K$ determines $\rho$ up to semisimplification.
In other words, we have an injection from the collection of degree $n$ semisimple $\ell$-adic representations of $G_K$ unramified outside $S$ to $\operatorname{Conj}_\text{ss}(\operatorname{GL}_n(\overline{\mathbb{Q}_{\ell}}))^{S_K \setminus S}$, where $\operatorname{Conj}_\text{ss}$ denotes the set of semisimple conjugacy classes (in a given algebraic group).
Each automorphic representation $\pi$ of $\operatorname{GL}_n(\mathbb{A}_K)$ splits as a restricted tensor product $\prod_{\mathfrak{p} \in S_K} \pi_{\mathfrak{p}}$, where $\pi_{\mathfrak{p}}$ is an irreducible representation of $\operatorname{GL}_n(K_{\mathfrak{p}})$. The definition of an automorphic representation ensures that this representation has a $\operatorname{GL}_n(\mathcal{O}_{\mathfrak{p}})$-fixed vector for almost all $\mathfrak{p}$, i.e., is unramified. Such representations are classified by the Satake isomorphism; they are given by semisimple conjugacy classes in $\operatorname{GL}_n(\mathbb{C})$. Furthermore, if $\pi$ is cuspidal, the collection of these conjugacy classes for almost all $\mathfrak{p}$ determines $\pi$ by the strong multiplicity one theorem.
In other words, we have an injection from the collection of cuspidal automorphic representations of $\operatorname{GL}_n(\mathbb{A}_K)$ unramified outside $S$ to $\operatorname{Conj}_\text{ss}(\operatorname{GL}_n(\mathbb{C}))^{S_K \setminus S}$.
Choosing an isomorphism $\overline{\mathbb{Q}_{\ell}} \cong \mathbb{C}$, we get a bijection
$$
\operatorname{Conj}_\text{ss}(\operatorname{GL}_n(\overline{\mathbb{Q}_{\ell}}))^{S_K \setminus S}
\cong
\operatorname{Conj}_\text{ss}(\operatorname{GL}_n(\mathbb{C}))^{S_K \setminus S}.
$$
In other words, a Galois representation gives you a collection of local data that is exactly the kind that uniquely determines a cuspidal automorphic representation, and an automorphic representation gives you a collection of local data that is exactly the kind that uniquely determines a Galois representation. The Langlands conjectures equate two different notions of this local data fitting together into a global object.
More precisely, the condition of fitting together into a representation of $G_K$ is a global condition on the former, and the condition of fitting together into an automorphic representation is a global condition on the latter. The Langlands conjecture essentially says that these two conditions are equivalent. (More precisely, it says that irreducible Galois representations that are de Rham at all places above $\ell$ should correspond to cuspidal automorphic representations that are “algebraic” at the infinite places.)
The idea of two different globality conditions being the same already sounds nice, but why should we really believe they correspond? Well, class field theory already tells us they do for $n=1$. That's pretty strong. But they are already known to correspond for many cases of $n>1$. For a more classical example, Eichler–Shimura says that an automorphic representation for $n=2$ with a particular condition at the infinite place (corresponding to holomorphic weight $2$) gives rise to a Galois representation compatibly with the above procedure.
Here's another way to see how the global conditions are similar. On the one hand, an element of $\operatorname{Conj}_\text{ss}(\operatorname{GL}_n(\overline{\mathbb{Q}_{\ell}}))^{S_K \setminus S}$ gives rise to a representation of the free product $\bigstar_{\mathfrak{p} \in S_K \setminus S} G_{\mathfrak{p}}$. Choosing lifts of each $\mathfrak{p}$ to $\overline{K}$, we get an epimorphism of topological groups $\bigstar_{\mathfrak{p} \in S_K \setminus S} G_{\mathfrak{p}} \to G_K$. The global condition is something like invariance under the kernel of this epimorphism. On the other hand, an element of $\operatorname{Conj}_\text{ss}(\mathrm{GL}_n(\mathbb{C}))^{S_K \setminus S}$ gives rise to a representation of $\prod_{\mathfrak{p} \in S_K \setminus S} \operatorname{GL}_n(K_{\mathfrak{p}})$, which we can restrict to the restricted direct product. The condition of automorphicity is something like invariance under $\operatorname{GL}_n(K)$.
