The main idea of the proof is the following: the 1-dimensional representations $\{\pi_{1}^{ab}(X) \rightarrow \bar{\mathbb{Q}}_{l}^{*}\}$ are in 1-1 correspondence with 1-dim local systems $L$ on $X$, on the other hand 1-dimensional representations $\{Pic_{X} \rightarrow \bar{\mathbb{Q}}_{l}^{*}\}$ are in 1-1 correspondence with 1-dimensional local systems $A_{L}$ on $Pic_{X}$ together with a rigidification, i.e. a fixed isomorphism $A_{L}|_{0} \cong \bar{\mathbb{Q}}_{l}$ (this is the famous faisceaux-fonctions correspondence of Grothendieck). Now consider the d-symmetric product $X^{(d)}$ of $X$ (which is just the effective divisors of degree $d$ on $X$) which maps in an obvious way to $Pic_{X}^{d}$, the degree $d$ component of the Picard. If there is given a local system $L$ on $X$ then we can produce a local system $L^{(d)}$ on $X^{(d)}$ by having this local system fibres $\bigotimes_{i} Sym^{d_{i}}(L_{x_{i}})$ over a point $\sum_{i} d_{i}x_{i}$. Now by Riemann-Roch if the degree $d$ is greater than $2g(X)-2$ then it follows that this map has fibre over a degree-$d$ line bundle $\mathfrak{L}$ the $d-g(X)$-dimensional projective space $\mathbb{P}(H^{0}(X,\mathfrak{L}))$. As projective spaces are simply connected, it follows that this locally constant sheaf $L^{(d)}$ is actually constant on these fibres, so descends to a local system $A_{L}$ on $Pic_{X}^{d}$. There is also a way to extend these construction to the remaining components using the natural action $X \times Pic \rightarrow Pic$ given by $(x,L) \mapsto L(x)$. It is the idea of the proof, which is actually of Deligne!

As what the references concerns: there is a paper of Laumon: Faisceaux automorphes lies aux series Eisenstein, where he discusses this proof of Deligne. Also here http://www.cims.nyu.edu/~tschinke/publications.html the 3rd book (Mathematisches Institut, Seminars 2003/04, Universitätsverlag Göttingen, (2004) ) from page 145, but it is in german. And also the quoted paper of Frenkel is very good.

Algebraic groups and class fields. $\endgroup$