My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T_p$ in terms of the geometric Frobenius map. Specifically, $T_p = Frob + Ver$ for Frobenius map $Frob$ and it's transpose $Ver$.
However, Wikipedia states that "the Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators".
I have no idea how that statement comes from the above definition of the congruence. How does such a decomposition of Hecke operators have anything to with its eigenvalues or L-functions of a modular curve?
EDIT: Will's amazing answer gives the connection between L-functions and eigenvalues, but how is the local L-function of a modular curve restated explicitly using Eichler-Shimura?