Intuition behind the Eichler-Shimura relation?

(1) Short answer to first question: $T_p$ is about $p$-isogenies, and in char. $p$ there is a canonical $p$-isogeny, namely Frobenius.

Details:

The Hecke correspondence $T_p$ has the following definition, in modular terms: Let $(E,C)$ be a point of $X_0(N)$, i.e. a modular curve together with a cyclic subgroup of order $N$. Now $T_p$ (for $p$ not dividing $N$) is a correspondence (multi-valued function) which maps $(E,C)$ to $\sum_D (E/D, (C+D)/D)$, where $D$ runs over all subgroups of $E$ of degree $p$. (There are $p+1$ of these.)

Here is another way to write this, which will work better in char. $p$: map $(E,C)$ to $\sum_{\phi:E \rightarrow E'}(E',\phi(C)),$ where the sum is over all degree $p$ isogenies $\phi:E\rightarrow E'.$ Giving a degree $p$ isogeny in char. 0 is the same as choosing a subgroup of order $D$ of $E$ (its kernel), but in char. $p$ the kernel of an isogeny can be a subgroup scheme which is non-reduced, and so has no points, and hence can't be described simply in terms of subgroups of points. Thus this latter description is the better one to use to compute the reduction of the correspondence $T_p$ mod $p$.

Now if $E$ is an elliptic curve in char. $p$, any $p$-isogeny $E \to E'$ is either Frobenius $Fr$, or the dual isogeny to Frobenius (often called Vershiebung). Now Frobenius takes an elliptic curve $E$ with $j$-invariant $j$ to the elliptic curve $E^{(p)}$ with $j$-invariant $j^p$. So the correspondence on $X_0(N)$ in char. $p$ which maps $(E,C)$ to $(E^{(p)}, Fr(C))$ is itself the Frobenius correspondence on $X_0(N)$. And the correspondence which maps $(E,C)$ to its image under the dual to Frobenius is the transpose to Frobenius (domain and codomain are switched). Since there are no other $p$-isogenies in char. $p$ we see that $T_p$ mod $p = Fr + Fr'$ as correspondences on $X_0(N)$ in char. $p$; this is the Eichler--Shimura relation.

(2) Note that only weight 2 eigenforms with rational Hecke eigenvalues give elliptic curves; more general eigenforms give abelian varieties.

An easy computation shows that if $f$ is a Hecke eigenform, than the $L$-funcion $L(f,s)$, obtained by Mellin transform, has a degree 2 Euler product. A more conceptual answer would probably involve describing how automorphic representations factor as a tensor product of local factors, but that it a very different topic from Eichler--Shimura, and I won't say more here.

Let me highlight some issues that Emerton doesn't:

1) you seem to hint that you don't know that modular forms can be viewed as a product of a bunch of local terms. So there is an adelic story, where GL_2(R)/GL_2(Z) gets replaced by GL_2(adeles)/GL_2(Q), and in that story Hecke operators are entirely local objects, each with its own local Euler factor on an eigenform etc etc. Maybe if you knew this story some stuff would be clearer---e.g. a modular form really does have a "local factor at p" for p a prime, but it's an infinite-dimensional representation of GL_2(Q_p)!

2) the "big picture" story of Eichler-Shimura, as I'm sure I've typed into this forum before somewhere, is that Langlands, a long time ago, made conjectures about how the cohomology of a very general class of Shimura varieties can be completely explained using automorphic forms, but the conjectures in full generality are very difficult to explain and have many subtleties (coming from endoscopy, non-compactness at infinity, multiplicity issues and so on). For modular curves the conjectures boil down to the statement that, vaguely speaking, the Tate module of the Jacobian of a compact modular curve X_0(N) should break up into 2-dimensional pieces each explained by an eigenform of weight 2 and level N. But there is a lot of stuff secretly built in there: you're using X_0(N) instead of Y_0(N), you're assuming "multiplicity 1" holds for cusp forms on GL_2, which is a theorem of Jacquet and Langlands, and so on. Once you take all this on board, the precise relation between the pieces and the modular forms is given to you by Eichler-Shimura. If you look at it in this way you can start to guess what e.g. the H^3 of the Siegel modular variety parametrising princ polarized abelian surfaces might look like, but your guess might be wrong, because now endoscopy and failure of multiplicity 1 and issues involving compactification really start rearing their ugly heads in a less trivial way. Somehow there are hundreds of pages of Corvallis devoted to this sort of thing, and one thing Clozel, Harris, Taylor and their collaborators have done in the last few years is to completely sort these issues out in the case of unitary groups when the Shimura variety is compact. Langlands' conjectures are stronger than, but imply, a generalised Eichler-Shimura correspondence in this setting, and probably Matt's approach involving considering correspondences in char p will also give some insights into what the precise statement should be. But my feeling is that it is genuinely hard to explain the big picture without a lot of work the moment one leaves GL_2. If you want to read about another case other than classical modular forms, perhaps the next easiest case is Hilbert modular forms. Good luck!

3) A nice place to read about Eichler-Shimura for GL_2/Q is the appendix by Conrad to "lectures on Serre's Conjectures" by Ribet-Stein.