On wikipedia (here) I have read the following:
Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at $−1$ i.e. $\zeta(-1) = -1/12$, the fact that the abelianization of $SL(2,\mathbb{Z})$ has twelve elements, and even the properties of lattice polygons.
I know the dimension formula and the weight formula for modular forms as well as the proof of the weight formula via the residue theorem, but I do not understand how this relates to any of the three concepts mentioned in the quote above. Can anybody tell me more?
Regarding the question in the title I must admit that, for me personally, "why" questions are always very hard to answer so I'll simply give some evidence that the (non-) existence of cusp forms can be viewed as a geometric phenomenon. What I am about to relate is very well known and details can be found in many places but I must say that I found the treatment offered in R. Hain's, LECTURES ON MODULI SPACES OF ELLIPTIC CURVES to be particularly instructive.
Let $\mathcal{M}_{1,1}=\operatorname{SL}_2(\mathbb Z)\backslash\!\!\backslash \mathfrak{H}$ be the moduli space of complex elliptic curves, where $\operatorname{SL}_2(\mathbb Z)$ acts on the upper half-plane $\mathfrak{H}$ in the usual way through Möbius transformations. Warning: this is not a complex manifold, rather a more complicated object called a complex orbifold. Nevertheless, all the algebraic invariants that can be associated to complex manifolds (such as fundamental group, homology groups, et cetera) can also be associated to $\mathcal{M}_{1,1}$. In particular, there is a good notion of (holomorphic) line bundle over $\mathcal{M}_{1,1}$ and, as per usual, we denote by $\operatorname{Pic}(\mathcal{M}_{1,1})$ the abelian group of isomorphism classes of such.
Now for our purposes the key statement is that:
$$ \operatorname{Pic}(\mathcal{M}_{1,1}) \cong \mathbb Z/12\mathbb Z $$
How does this relate to the (non-) existence of cusp forms? The point is that there exists a line bundle $\mathcal{L}$ on $\mathcal{M}_{1,1}$ such that global sections of its $k$-th tensor power $\mathcal{L}^{\otimes k}$, for $k \in \mathbb Z$, are so-called ``weakly holomorphic'' (:=holomorphic on $\mathfrak{H}$ and meromorphic at the cusp $\infty$) modular forms of weight $k$. On the other hand, a line bundle is trivial if and only if there exists a nowhere vanishing global section and since $\mathcal{L}^{\otimes 12}$ is necessarily trivial by the above result, there exists a weakly holomorphic modular form of weight $12$ that vanishes nowhere on $\mathfrak{H}$. The $k/12$ formula then forces it to vanish at $\infty$ as well and that is your cusp form.
In fact the class of $\mathcal{L}$ even generates $\operatorname{Pic}(\mathcal{M}_{1,1})$ so that $12$ is indeed the smallest weight in which a non-zero cusp form can exist.
A final comment, since you mention $\zeta(-1)$ in this context: That is actually equal to the Euler characteristic $\chi(\mathcal{M}_{1,1})$ of $\mathcal{M}_{1,1}$. More generally Harer and Zagier proved that $$ \chi(\mathcal{M}_g)=\zeta(1-2g), $$ for all $g\geq 2$, where $\mathcal{M}_g$ denotes the orbifold of compact Riemann surfaces of genus $g$.
PS: I would have liked to vote up this question but since at the time of my answer it had exactly $12$ upvotes already I light-heartedly decided against that.