That's how I see it. Please correct me if I am wrong:

Induce the trivial representation from a lattice $\Gamma$, e.g. $\mathrm{SL}_2( \mathbb{Z})$, to the group $\mathrm{SL}_2(\mathbb{R})$.

Then consider the $\mathrm{SO}_2(\mathbb{R})$-invariant subspace.(weights are actually associated to one dimensional representations of SO_2)

This representation is isomorphic to the representation $L^2(D,dz)$, you construct above.

How does the Laplace Beltrami operator enter the picture...
It is the descent of the Casimir operator to this space, which is the generator of all invariant differential operators (= center of the universal envelopping algebra of \mathfrak{sl}_2(\mathbb{R})). So intuitevely the Casimir operator captures the $G$ structure on the Hilbert space.

Why did we became interested in such constructions?
Maass discovered that the Mellin transforms of Dedekind zeta function associated to a quadratic real fields are Maass forms.

Now, since all interesting $L$ functions (ass. to Galois repr., elliptic curves,...) are conjectured to be associated to some representation to some group, it seems worthwhile:
1. To study if and how they are associated ... (e.g., Taniyama-Shimura conjecture)
2. To study their properties on either side and conclude about the other...

Mappings between groups and comparing their L-functions give also nice information about them (functoriality), e.g. they generalized Ramanujan conjecture would be implied by certain "functoriality" conjectures betweens general linear groups .

Perhaps I should conclude that with the adeles $\mathbb{A}$ the better picture is
$$ \mathcal{L}^{2} ( \mathrm{SL}_2(\mathbb{Q}) \backslash \mathrm{SL}_2(\mathbb{A}))^{\mathrm{K}(m)\mathrm{SO}(2)} \cong \mathcal{L}^2(\Gamma(m) \backslash \mathbb{H}),$$

where $\mathrm{K}(m)$ is the product over $\mathrm{K}_p(m)$, the group of elements in $\mathrm{SL}_2( \mathbb{Z}_p)$ which are the identy modulo $m$. This picture contains the Hecke operators more naturally...