Intuition for Zagier's theorem for $\zeta_K(2)$ In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})...A(x_{v,s})$$
where
$$A(x)=\int^x_0 \frac{1}{1+t^2}\log \frac{4}{1+t^2}dt$$
In this sense, the result is conjectured to hold for $2<s\in \mathbb{N}$, with the $A(x)$ replaced by more complicated functions $A_m(x)$
This might seem rather unenlightening, but we can also state Zagier's result like this:

$$\zeta_K(2)=\text{the volume of a hyperbolic manifold}$$

This amazing fact doesn't seem to have a direct analogue for $\zeta_K(2m)$ with $m \neq 1$


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*What I'd like to know is if there is any big picture explanation for the appearance of hyperbolic manifolds in this context.


Zagier's calculation is quite geometrical, but as far as I understand gives no clear explanation of "what the manifold is doing here".
Don Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions (1986)
 A: Too long for a comment and not sure whether this counts as a big picture, but anyway:
There is a general formula for the covolume of S-arithmetic lattices in symmetric spaces, you find it in Prasad's original paper or in Remy's Bourbaki talk.
It is a product of several factors, one of them (the so-called Euler factor) contributes the $\zeta_K(2)$ for certain hyperbolic 3-manifolds (those considered in Zagier's paper.)
I'm not an expert in this formula and it is not easy to see what it actually gives. Apparently for other lattices one seems to get other zeta- or L-functions, but apparently there is no lattice (at least in hyperbolic  3-space) which gives $\zeta_K(2m)$.
A: As ThiKu mentions, the connection between $\zeta_K(2)$ and hyperbolic manifolds is that the volume formula for arithmetic hyperbolic manifolds is given by an explicit formula involving $\zeta_K(2)$. This formula is due to Borel in the case of arithmetic manifolds arising from quaternion algebras. A more general formula was later given by Prasad.
In order to get some intuition for general hyperbolic surfaces or $3$-manifolds, it might help to start by recalling the modular surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ where $\mathfrak h_2$ denotes the hyperbolic plane. It is known that $\mathrm{vol}(\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2)=\frac{2}{\pi}\zeta(2)=\frac{\pi}{3}$. (See for instance Section 2 of these notes by Garret or these notes by Venkatesh.) So in this case we see the appearance of a zeta value in the context of the volume of a hyperbolic surface.
Here is an admittedly circuitous way to construct the hyperbolic surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ that will provide some intuition for what happens more generally. Consider the quaternion algebra $B=\mathrm{M}_2(\mathbb Q)$ and the maximal order $\mathcal O=\mathrm{M}_2(\mathbb Z)$ inside this quaternion algebra. Let $\mathcal O^1=\mathrm{SL}_2(\mathbb Z)$ denote the multiplicative group of elements of $\mathcal O$ having determinant $1$ and let $\Gamma$ denote the image of $\mathcal O^1$ inside $\mathrm{SL}_2(\mathbb R)$ where we make use of the map $B\hookrightarrow B\otimes_{\mathbb Q} \mathbb R\cong \mathrm{M}_2(\mathbb R)$. The group $\Gamma$ is a discrete subgroup of $\mathrm{SL}_2(\mathbb R)$ which has finite volume, and as $\mathrm{SL}_2(\mathbb R)$ is the group of orientation preserving isometries of $\mathfrak h_2$, we get our quotient surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$.
The connection which Zagier makes between zeta values and manifolds ultimately arises from a "number field" analog of the above. 
(A very interesting fact that is not directly related to your question by which I cannot resist mentioning is that many aspects of the geometry of manifolds defined in this sort of arithmetic manner (i.e., via the construction given below) are directly related to quantities of number theoretic interest. For instance, the lengths of closed geodesics on $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ correspond to regulators of real quadratic fields and their multiplicity to the class number of the associated real quadratic field. This connection generalizes to manifolds constructed from number fields other than $\mathbb Q$ as well.)
Let $K$ be a number field with $r_1$ real places and $r_2$ complex places and $B$ be a quaternion algebra over $K$ which is not totally definite. Let $s$ denote the number of real places of $K$ that split in $B$. Thus $$B\otimes_{\mathbb Q} \mathbb R\cong \mathbb H^{r_1-s}\times \mathrm{M}_2(\mathbb R)^{s} \times \mathrm{M}_2(\mathbb C)^{r_2}.$$ Let $\mathcal O$ be a maximal order of $B$ and $\mathcal O^1$ the multiplicative subgroup consisting of elements with reduced norm $1$. (If $B$ was a matrix algebra then the reduced norm and determinant would coincide.) Let $\Gamma_\mathcal{O}$ denote the image of $\mathcal{O}^1$ in the group $$G_{s,r_2}=\mathrm{SL}_2(\mathbb R)^{s}\times \mathrm{SL}_2(\mathbb C)^{r_2}.$$ The group $G_{s,r_2}$ is the group of orientation preserving isometries of $\mathfrak h_2^{s}\times \mathfrak h_3^{r_2}$ which preserves factors. Here $\mathfrak h_3$ is hyperbolic $3$-space. The group $\Gamma_\mathcal O$ is a discrete subgroup of $G_{s,r_2}$ which is cocompact if $B$ is a division algebra and has covolume given by the formula $$\mathrm{vol}(\Gamma_\mathcal O\backslash \mathfrak h_2^{s}\times \mathfrak h_3^{r_2})=\frac{2(4\pi)^sd_K^{3/2}\zeta_K(2)}{(4\pi^2)^{r_1}(8\pi^2)^{r_2}}\prod_{\mathfrak p\in\mathrm{Ram}_f(B)}\left(N(\mathfrak p)-1\right),$$ where $d_K$ is the absolute value of the discriminant of $K$ and $\mathrm{Ram}_f(B)$ is the set of finite primes of $K$ which ramify in $B$. This formula is proven in Section 7.3 of Borel's paper. In particular note the appearance of $\zeta_K(2)$ in the formula.
The following two special cases of all of this are worth noting:


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*When $s=1$ and $r_2=0$ the manifolds (or orbifolds) constructed are arithmetic hyperbolic surfaces.

*When $s=0$ and $r_2=1$ the manifolds (or orbifolds) constructed are arithmetic hyperbolic $3$-manifolds (or $3$-orbifolds).


Finally, note that more general zeta values like $\zeta_K(2m)$ do appear in the volume formulas for different types of arithmetic manifolds. This is already the case for higher dimensional arithmetic hyperbolic manifolds. See for instance some of the formulas in this paper by Belolipetsky and Emery. (As was mentioned above, these zeta values arise when one works out the relevant case of Prasad's general volume formula.)
A: There is a K-theory reformulation of Zagier's theorem which explains both the appearance of the integrals $A(x)$ as well as the appearance of hyperbolic geometry. I will try to give a sketch what is behind the relation of the zeta-value with volumes of hyperbolic simplices. 
First, due to the work of Borel, there is a relation between zeta-values and regulators for K-theory. For a number field $K$, the Borel regulator embeds $K_{2n-1}(\mathcal{O}_K)$  as a lattice in some $\mathbb{R}$-vector space (later reinterpreted by Beilinson as Deligne-cohomology group $H^1_{\mathcal{D}}(\operatorname{Spec} K\otimes\mathbb{C},\mathbb{R}(n))$). The covolume of the lattice is, up to multiplication by a power of $\pi$ and an algebraic number, identified with the zeta-value $\zeta_K(-n+1)$. So, Borel's theorem states: The zeta-value is the covolume of the regulator embedding $K_3(\mathcal{O}_K)\to\mathbb{R}^{r_2}$.
In a second step, there is a relation between K-theoretic regulators and dilogarithms (following the work of Bloch, Suslin, Dupont-Sah...). Recall that the pre-Bloch group $\mathcal{P}(K)$ is given by generators $x\in K^\times\setminus\{1\}$ subject to the five-term relation $$[x]-[y]+[y/x]-[(1-x^{-1})/(1-y^{-1})]+[(1-x)/(1-y)]$$ for $x, y\in K^\times\setminus\{1\}$. The Bloch group $\mathcal{B}(K)$ is the kernel of the map
$$
\mathcal{P}(K)\to\bigwedge^2 K^\times:[x]\mapsto x\wedge(1-x).
$$
The five-term relation is a version of the functional equation for the dilogarithm. Hence, by definition, the single-valued real-analytic version of the dilogarithm yields a function mapping the Bloch group into $\mathbb{R}^{r_2}$ by taking a class $[x]$ to the values of the dilogarithm under the various complex embeddings $K\to\mathbb{C}$. The Bloch group can (up to 2-torsion) be identified with the indecomposable quotient of $K_3(\mathcal{O}_K)$. The important thing is that under this identification, the regulator on $K_3$ is mapped to the dilogarithm function on the Bloch group. (There are some factors, algebraic numbers and powers of $\pi$, but this does not cause problems for the identification.)
It is mentioned in Zagier's paper that $A(x)$ is essentially the dilogarithm. The combination of the above two steps yields the reformulation of Zagier's theorem: Expressing the regulator embedding by dilogarithms, the zeta-value is a determinant of dilogarithms of elements of $K$.
In a third step, the relation to hyperbolic geometry can now be obtained from the Bloch group. Via the cross-ratio, the pre-Bloch group is isomorphic to the third homology of the complex of points on $\mathbb{P}^1(K)$ (with the omit-one-point-differential) modulo the diagonal action of $GL_2(K)$. This reinterprets elements of the Bloch group as ordered $4$-tuples of points on $\mathbb{P}^1(K)$. Now four points on $\mathbb{CP}^1$ span a simplex in $\mathbb{H}^3\cup\partial\mathbb{H}^3$. One can talk about hyperbolic $3$-polytopes up to scissors congruence. It is possible to prove that the group of scissors congruence classes in $\mathbb{H}^3$ is isomorphic to the group of scissors congruence classes of polytopes in $\mathbb{H}^3\cup \partial\mathbb{H}^3$ whose vertices lie on $\partial\mathbb{H}^3$ (this is done in the book "Scissors congruences, group homology and characteristic classes" by J.-L. Dupont). 
We can now explain the link between the Bloch group and hyperbolic geometry by: $\partial\mathbb{H}^3\cong \mathbb{CP}^1$. More precisely, the Bloch group is a group of scissors congruence classes of hyperbolic $3$-polytopes with vertices on $\partial\mathbb{H}^3$.
Combining the above three steps means that $\zeta_K(2)$ can be expressed as a combination of volumes of hyperbolic simplices. This statement actually has a conjectural generalization for all $\zeta_K(n)$. Zagier formulated the corresponding conjecture in "Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields". There are analogues of the Bloch group which conjecturally can be identified with higher K-groups $K_{2n-1}(\mathcal{O}_K)_{\mathbb{Q}}$, such that the regulator on the K-group can be expressed in terms of $n$-logarithms. If true, this conjecture would allow to express $\zeta_K(n)$ as a determinant of $n$-logarithms, or equivalently as a linear combination of volumes of hyperbolic $(2n-1)$-simplices with vertices on the boundary. The trilogarithm case of Zagier's conjecture was proved by Goncharov, see


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*A. Goncharov. Volumes of hyperbolic manifolds and mixed Tate motives. J. Amer. Math. Soc. 12 (1999), 569-618. 


As a final note, the relation between polylogarithms and volumes of hyperbolic simplices goes back to Lobachevskiy, Schläfli,... 
