Two other applications where automorphic forms (again: only some of them, so it is more a motivation for modular forms or for Maass forms than for automorphic forms) arise naturally are provided in Kowalski's article, Classical Automorphic Forms, in Bernstein-Gelbart volume.

Kloosterman Sums. Estimating cancellations arising in Kloosterman sums, for which best results are given by automorphic methods. For instance, it is used to address the problem of estimating the size of $L$-functions on the critical line, and more precisely associated moments $$\int_{-T}^T \left|L\left(\frac12+it, \pi\right)\right|^k dt$$

The strong implications of good bounds on those moments (e.g. Lindelöf) are a motivation for addressing the problem, and since automorphic forms are for now among the best way to address it, it is a motivation to automorphic forms.

Elliptic curves. Modular functions can be seen as meromorphic functions homogeneous on lattices, and for instance de Weierstrass function $\mathcal{P}$ give invariants determining the elliptic curve.

Again, for the reason stated above, these fail to totally answer my question (even for audience (D) I believe, for it is only a partial answer, or better: an answer to part of the question).