What motivations for automorphic forms?

To give a brief answer, which I think applies to all audiences, and I hope is not too "elementary" for you (I'm not attempting to give details, which of course need to be specialized for the intended audience, and I'm not entirely sure what you're looking for):

Automorphic forms provide analytic ways to study solutions to equations in integers or rational numbers.

Ex 1 How many ways can one represent $n$ as a sum of $k$ squares, i.e., how many integer solutions does $x_1^2 + \cdots + x_k^2 = n$ have?

If we call this number $r_{k}(n)$, then the Fourier series $$ f_{k}(z) = \sum_{n=0}^\infty r_{2k}(n) e^{2\pi i nz}$$ is a modular form of weight $k/2$ and level $4$. Studying the analytic behaviour of this function allows one to get asymptotics for $r_{2k}(n)$, and more precise studies of Fourier coefficients of modular forms allows one to get actual formulas for $r_{2k}(n)$, at least for certain values of $k$.

The study of quadratic forms motivated much of the arithmetic study of modular forms, of which automorphic forms generalize. Asking more complicated questions leads to other kinds of automorphic forms (e.g., what quadratic forms represent what forms leads to Siegel modular forms).

Ex 2 What integers are sums of two rational cubes, i.e., when does $x^3+y^3=n$ have a solution over $\mathbb Q$?

This is a special case of asking when elliptic curves have infinitely many rational points, which is related to modular forms via $L$-functions and the Birch and Swinnerton-Dyer conjecture. Namely one wants to study zeroes of $L$-functions, but one needs to pass to modular forms/automorphic representations to get $L$-value formulas. Fermat's last theorem is of course more famous, but I like this problem a lot.

The specific issue of what automorphic forms on bigger groups than $GL(2)$ over $\mathbb Q$ (for example) may tell us about automorphic forms (and L-functions) for $GL(2,\mathbb Q)$ or $GL(1,k)$ for number fields $k$ does have at least a few good answers. First, about 1960 and a little before, Klingen's proof that zeta functions of totally real number fields $k$ (and L-functions of totally even characters on such fields) have good special values at positive even integers used the idea of pulling back holomorphic Hilbert modular Eisenstein series to elliptic modular forms. (I heard G. Shimura lecture on this c. 1975, and it was quite striking.)

Another example: already in the 1960s, J.-P. Serre and others saw that holomorphic-ness of symmetric-power L-functions for $GL(2)$ holomorphic modular forms would prove Ramanujan-conjecture-type results. How to prove that holomorphy? By finding an integral representation of such L-functions, and using that. This has met with varying degrees of limited success, e.g., in papers of H. Kim and F. Shahidi.

The previous example was grounded in the general pattern of Langlands-Shahidi treatment of L-functions in terms of constant terms of cuspidal-data Eisenstein series on (necessarily larger) reductive groups. The specifica cases where various Levi-Malcev components of parabolics were products of $GL2$'s or $SL2$'s produced several "higher" L-functions for $GL2$.

As variation on that, already in the Budapest conference in 1971, Piatetski-Shapiro observed that (what we often nowadays call) Gelfand pairs could produce Euler produces via integral representations (usually involving Eisenstein series). Various success-examples of this idea included work of PiatetskiShapiro-Rallis, Shimura, myself, M. Harris, S. Kudla, various collaborations among these people, and several others, beginning in the late 1970s. E.g., I was fortunate enough to stumble upon an integral representation for triple tensor product L-functions for $GL2$ in terms of an integral representation against Siegel Eisenstein series on $Sp(3)$ (or $Sp(6)$, if one prefers). M. Harris and S. Kudla found another such integral representation that covered special value results in the "other range" (in terms of P. Deligne's conjectures).

In yet other terms, Jacquet-Lapid-Rogawski (and several others) have demonstrated that a variety of L-functions appear as periods of Eisenstein series on "larger" reductive groups. (One novelty is using relative trace formulas to exhibit Euler products when the simpler "Gelfand pair" idea is not quite sufficient.)

Here's an answer for audiences (C), (D) about class field theory:

Class field theory provides a way classify abelian extensions of number fields $F$. In the case of $F=\mathbb Q$, this is answered by the Kronecker-Weber theorem, which says any abelian extension is contained in $\mathbb Q(e^{2\pi i/n})$ for some $n$. The fields $\mathbb Q(e^{2\pi i/n})$ are the ring class fields of $\mathbb Q$. To generalize this to other number fields $F$, one would like an analogue of the extensions $\mathbb Q(e^{2\pi i/n})$. If $F$ is imaginary quadratic, these are extensions obtained by adjoint special values of a certain transcendental function called the $j$-invariant, which is a quotient of classical modular forms.

To study extensions to more general number fields $F$, Hilbert was led to consider a generalization of classical modular forms to what are now known as Hilbert modular forms, again special cases of automorphic forms. Then, at least in some cases, one can use automorphic forms to construct certain special functions whose special values can be used to construct abelian extensions of $F$.

Much more generally, but also much more vague (at least at present), there is the Langlands program. (Here I will be quite vague on details---feel free to edit more in or make another answer about this.) This has many aspects, but the one related to class field theory is the point of view that class field theory can be regarded as a correspondence between 1-dimensional Galois representations (representations of Gal($\bar F/F$)) and automorphic representations of GL(1) (Hecke characters, like Dirichlet characters). At the heart of the Langlands program is a conjectural correspondence between $n$-dimensional Galois representations and "automorphic representations of GL($n$)". This is often called nonabelian class field theory, because it should provide a way to describe the non-abelian extensions of $F$.