For a number field $K$ with Dirichlet zeta function $\zeta_K(s), $ we have the class number formula: $$\lim_{s \to 1^+} (s-1)\zeta_K(s) =\cfrac{2^{r_1}(2\pi)^{r_2}Rh}{m\sqrt{(|\Delta)|}}$$ where $r_1$ is the number of real embeddings of the field, $r_2$ for complex -nonconjugate- ones; $R$ for the regulator of $K, h$ is the class number of $K$, $m$ is the number of roots of unity in $K$ and $\Delta$ is the discriminant of $K$.
Intuitively, how can we interpret the regulator $R$ of $K$ and what can we say about where does $\pi$ come from ?
(NB : This was meant to be a comment, not an answer, but it became rapidly too lengthy)
It seems to me that the point is not the appearence of $\pi$ in the class number formula (after all, this could be considered as a technical by-product of the proof), but the deeply mysterious way in which this formula binds together an algebraic object (the class group) and an analytic object (the $\zeta$- function) by means of a transcendental determinant (the regulator). I don't quite agree with @Mathmo123 that Tate's thesis really explained why this happens. Actually, Tate resolutely adopted the global-local point of view by defining generalized $\zeta$- functions as integrals over the idèle group of certain weight functions, which allowed him to apply the full force of abstract Fourier analysis in locally compact abelian groups to establish at one stroke an analytic continuation and a functional equation. This idelic approach illustrates the power of the global-local principle in number theory, which consists in putting the the $p$-adic worlds on an equal footing with the archimedean world . But it remains to explain why this principle works so well.
Coming back to the OP question, it is natural to turn it around and wonder why the class number $h$ pops up in a formula giving the residue of $\zeta_F$ at $s=1$. Actually the powers of $\pi$ can be cancelled by applying the functional equation, which yields the formula $\zeta_F(0)^*=-Rh/w $ for the special value $\zeta_F(0)^* :=$ the first non zero coefficient in the Taylor expansion of $\zeta_F$ at $s=0$. Here $w$ is the order of the group of roots of unity contained in $F$. Not only the special value at $0$ looks simpler, but the rational number $h/w$ makes you want some of the same at all the negative values $s=-n, n\in \mathbf N$. Hints are given by the special values of the Riemann $\zeta$-function. It is classically known that : $\zeta(0)=-1/2, \zeta (1-2m)=-B_{2m}/2m$, where $B_k$ is the $k$-th Bernoulli number, $-2m$ is a simple zero, $\zeta(-2m)^*= (?)$ (the mystery is the same for $\zeta(2m+1)$). Number theorists have a mannerism : when facing a rational number, they inevitably ask whether the numerator and the denominator could be the orders of some finite groups . Astonishingly, this is the case. At the beginning of the 1970's, Lichtenbaum proposed the following conjecture (some jargon is inevitable here) : for any number field $F$, $\zeta_F(1-m)^*=\pm 2^? R_m\mid K_{2m-2}O_F\mid/\mid tors K_{2m-1}O_F\mid$ for any $m \ge 2$. Notations : the exponent (?) can be made precise; $R_m$ is the Borel-Beilinson regulator, an elaborate generalization of the Dedekind regulator $R$ (see below); the Quillen groups $K_{i}O_F$ are topological objects attached to the ring of integers $O_F$. For some more details, see e.g. [BK].
The Lichtenbaum conjecture is now a theorem when $F$ is an abelian field. But the proof was made possible only inside the (partly conjectural) framework of the Bloch-Kato conjectures on the special values of "motivic L-functions". The so called search for "motives" goes back to an original idea of Grothendieck, which one could find far-fetched but which I prefer to call "platonist". Remember Plato's "apologue of the cavern": we humans live in a cave, and the physical reality which we perceive consists in shadows cast on the walls by the sun in our backs; studying these shadows could occupy a lifetime, but to understand the true "reality", we must turn around and face the archetype which projects these shadows. Grothendieck applied this philosophical concept to algebraic/arithmetic geometry : around a given variety are floating a host of dissimilar cohomologies (Betti, de Rahm, étale...), which become isomorphic when passing to an algebraic closure, but such a passage destroys all the arithmetical properties we are interested in. Following Plato, Grothendieck suggested to look not at the shadows but at the archetype, the conjectural motivic cohomology, which would cast its shadows by means of regulator maps. this was achieved at the beginning of this century by Voevodsky, giving in particular a precise relationship beween K-theory and Galois/étale cohomology via Chern clas maps (=the regulator maps, whose determinants are the regulator numbers which appear in the special values).
To summarize : the blend of algebra-analysis-topology in the special values of $\zeta$ certainly remains mysterious, but it is natural, in that it actually reveals the profound unity of mathematics. There is a legend (?) about the last words of Hermite. On his death bed he would have said : "Now I'll be able to see Zeta face to face".
[BK] "The Bloch-Kato Conjecture for the Riemann Zeta Function", Proceedings of the 2012 Pune conference, edited by John Coates & al., London Math. Soc. LNS 418,2015