Why are Shimura varieties the "right" objects? So this is probably blasphemist to ask and I've resisted asking this for a while. Essentially my question is why are locally symmetric spaces/Shimura varieties the "right" object to study from a number theoretic perspective?
The merit of them, say from a Langlandian perspective, is for sure beyond debate, and certainly not my question. Specifically they are spaces with a Hecke-action (and even a Galois one in the Shimura case), such that we can assign to Hecke-eigenspaces of its cohomology a Galois representation (or at least conjecturally).
However, it could be, at least naively, that there exist other spaces with similiar number-theoretic benefits.
 A: This is a long comment. Number theorists have long been interested in modular functions and modular forms, which are functions on the complex upper half plane. Elliptic modular curves are the spaces on which these functions live. An important property of modular curves, which helps explain why they are of interest to number theorists, is that they have canonical models over number fields.
A modular curve is the quotient of a one-dimensional bounded symmetric domain (= hermitian symmetric domain) by a congruence subgroup. Remove the "one-dimensional", and you get a Shimura variety. Shimura varieties are known to have canonical models over number fields, and so have a reasonable claim to being the "right" generalization of modular curves.
From another perspective, a modular curve is a moduli variety for one-dimensional abelian varieties with additional structure. Remove the "one-dimensional" and you get a PEL Shimura variety. I suspect algebraic geometers would have considered these to be the "right" generalization if Shimura hadn't proved that Shimura curves (not of PEL-type) also have canonical models. Shimura varieties with integral weight are moduli variety for motives with additional structure, at least conjecturally.
Shimura varieties play two roles in the Langlands program, first as sources for Galois representations and second as a test of Langlands's idea that all L-functions arising from algebraic geometry are automorphic. For the first, PEL Shimura varieties suffice.  Much is known about the zeta functions of Shimura varieties and almost nothing about the zeta functions of other varieties (except for elliptic curves over $\mathbb{Q}$ and some modest generalizations).
Thus, Shimura varieties are certainly good objects for number theorists to study (but not the only good objects).
