Why is 12 the smallest weight for which a cusp forms exists

"Lattice Polygons and the Number 12" by Poonen and Rodriguez-Villegas relates the 12 in the Riemann-Roch theorem, to the 12 in the weight of the cusp form, to a property of convex lattice polygons. It's a nice read. You can find a copy of the paper here:

http://www.mimuw.edu.pl/~jarekw/SZKOLA/toric/LaticePolyAndNumb12.pdf

These facts are related via the geometric interpretation of cusp forms. Essentially, identifying cusp forms as differential forms on modular curves allows to establish the relation to the ${\rm SL}_2\mathbb{Z}$-action on the upper half-plane. The following is very sketchy, but I hope it properly outlines the main points.

The first step is to use the transformation behaviour to interpret cusp forms as certain holomorphic differential forms on a modular curve. Alternatively, cusp forms are viewed as global sections of a suitable line bundle on the compactification of ${\rm SL}_2\mathbb{Z}\backslash\mathbb{H}$. Then one can apply Riemann-Roch to compute the dimension of the space of cusp forms in terms of the Euler characteristic of the modular curve. A formula for dimensions of spaces of cusp forms for congruence subgroups can be found as Theorem 4.9 in Milne's lecture notes. Formulas also applicable to ${\rm SL}_2\mathbb{Z}$ are discussed in the following lecture notes these lecture notes. For ${\rm SL}_2\mathbb{Z}\backslash\mathbb{H}$, the formula for the Euler characteristic gives $-\frac{1}{12}$ because there is one point with isotropy $\mathbb{Z}/4\mathbb{Z}$ and one point with isotropy $\mathbb{Z}/6\mathbb{Z}$. (There is some subtlety here with the isotropy which leads to the Euler characteristic not being integral. One can also deal with this by considering congruence subgroups $\Gamma$ such that $\Gamma\backslash\mathbb{H}$ is a Riemann surface and apply some form of Riemann-Hurwitz to compare to ${\rm SL}_2\mathbb{Z}\backslash\mathbb{H}$.)

Now the Euler characteristic can be compared to the order of the abelianization. The fundamental domain for the ${\rm SL}_2\mathbb{Z}$-action retracts equivariantly onto one of its boundary segments, leading to an equivariant retraction of the upper half plane onto a tree. From the action of ${\rm SL}_2\mathbb{Z}$ on the tree we get the well-known amalgam decomposition $\mathbb{Z}/4\mathbb{Z}\ast_{\mathbb{Z}/2\mathbb{Z}}\mathbb{Z}/6\mathbb{Z}$, where the subgroups are the isotropy group from before. The natural map from this amalgamated product to $\mathbb{Z}/12\mathbb{Z}$ is in fact the abelianization -- hence we get a relation between the abelianization and the isotropy of the action on $\mathbb{H}$.

Now that we have the Euler characteristic, we can use the Gauss-Bonnet theorem (in a version for finite-volume hyperbolic surfaces) to relate volume and Euler characteristic $$ -{\rm vol}(M)=2\pi\chi(M) $$ This should rather be applied to ${\rm PSL}_2\mathbb{Z}$ because of the ubiquitous complication that the center (of order 2) in ${\rm SL}_2\mathbb{Z}$ fixes all of $\mathbb{H}$ -- the result is a factor 2. In the end, the volume for ${\rm SL}_2\mathbb{Z}\backslash\mathbb{H}$ turns out to be $\pi/3$.

Now we need to compare this with another computation of the volume related to the zeta values. There is an equality of volumes $$ {\rm Vol}({\rm SL}_2\mathbb{Z}\backslash\mathbb{H})\times {\rm Vol}({\rm SO}(2,\mathbb{R}))=2{\rm Vol}({\rm SL}(2,\mathbb{R})/{\rm SL}_2\mathbb{Z}). $$ The right-hand side is the Tamagawa volume which is $\zeta(2)$. The volume of ${\rm SO}(2,\mathbb{R})$ is $\pi$. The combination tells us that the volume of ${\rm SL}_2\mathbb{Z}\backslash\mathbb{H}$ is $2\zeta(2)/\pi$. Together with the functional equation for $\zeta$, we see that $-\zeta(-1)$ is the Euler characteristic of ${\rm SL}_2\mathbb{Z}\backslash\mathbb{H}$ which entered in the computation for the cusp forms. For the volume statements, see chapter 11 of MacLachlan-Reid: The arithmetic of hyperbolic 3-manifolds. The appearance of $\zeta(2)$ can be seen directly in the volume computation.