Why are Green functions involved in intersection theory?

The Green's function is used not to measure distances in the surface but to measure distances in the line bundle. A Green's function on $X_{\mathbb C}$ that blows up at $D$ can be used to measure sections of $\mathcal O(D)$. Indeed, if we represent a section of $\mathcal O(D)$ as a holomorphic function $f$ on $X_{\mathbb C}$ with poles at the points of $D$, and $g$ is a Green's function on $X_{\mathbb C}$ with poles at $D$, then $|f(x)|^2 e^{- |g(x)|}$ is a smooth nonnegative function that vanishes only where $f$ vanishes as a section of $\mathcal O(D)$ - i.e. where it lies in the image of the natural map $\mathcal O(D-P) \to \mathcal O(D)$. In fact it is easy to see that this is a Hermitian form on the space of sections. So the Green's function gives a metric on the line bundle.

We need a metric at $\infty$ in an Arakelov line bundle because we already have a metric (in the form of a $p$-adic valuation) at every other place - any section of $\mathcal O(D)$ on $X_{\mathbb Q_p}$ has a well-defined $p$-adic valuation at each point of $x\in X_{\mathbb Q_p}$, where a local generator of the line bundle at the reduction mod $p$ $\overline{x}\in X_{\mathbb F_p}$ is given absolute value $0$ - because such a generator is also a generator of the fiber of $\mathcal O(D)$ at $x$, this determines the valuation on every section.

It is a good idea to restrict first to understand the case of arithmetic surfaces, which is much easier than the general case. For example, you do not have to care about Green currents (only Green functions) or the *-product. Also, there is Lang's very readable book "Introduction to Arakelov Theory" treating this case.

Next, let me clarify some things. An Arakelov divisor on an arithmetic surface $X$ is a formal sum $$D=D_{\mathrm{fin}}+\sum_{\sigma}r_\sigma F_\sigma,$$ where $D_{\mathrm{fin}}$ is a classical Weil divisor on $X$, $F_\sigma$ is just a symbol standing for "a fibre at infinity" and the $r_\sigma$'s are real numbers and not Green functions.

The Arakelov-Green function associated to a Riemann surface $X_\sigma$ is indeed a function measuring the distance of two points on $X_\sigma$. To see this, let me recall its definition: $G_\sigma\colon X_\sigma\times X_\sigma\to \mathbb{R}_{\ge 0}$ is the unique function satisfying:

(i) $G_\sigma^2$ is $C^{\infty}$.

(ii) $G_\sigma(P,Q)$ is non-zero for $P\neq Q$. For fixed $Q_0\in X_\sigma$ the function $G_\sigma(P,Q_0)$ has a simple zero in $P=Q_0$.

(iii) If $P\neq Q$, its curvature is given by $\partial_P\overline{\partial_P}\log G_{\sigma}(P,Q)=\pi i \mu_\sigma(P)$.

(iv) It is normalized by $\int_{X_\sigma}\log G_\sigma(P,Q)\mu_\sigma(P)$.

Here, $\mu_\sigma$ is the canonical 1-1 form on $X_{\sigma}$ defined by $\mu_\sigma=\frac{i}{2g}\sum_{j=1}^g \omega_j\wedge\overline{\omega_j}$, where the $\omega_j$'s form an ON-basis of $H^0(X_{\sigma},\Omega_{X_{\sigma}}^1)$ with respect to the inner product $\langle \omega,\omega'\rangle=\frac{i}{2}\int_{X_\sigma}\omega\wedge\overline{\omega'}$. The property (ii) shows, that $G_\sigma(P,Q)$ measures the distance of $P$ and $Q$ on $X_\sigma$.

It remains to say, what is the connection of Arakelov divisors and the Arakelov-Green function. Instead of Arakelov divisors one can consider admissible metrized line bundles. A metrized line bundle is admissible if $\partial \overline{\partial}\log\|s\|_{\sigma}$ is a multiple of $\mu_{\sigma}$ for a non-vanishing local section $s$ of this line bundle. It turns out, that the notions of admissible metrized line bundles up to isomorphisms and Arakelov divisors up to principal Arakelov divisors are canonically equivalent.

To see this, we equip the line bundle $\mathcal{O}_{X_\sigma}(P)$ for a section $P\colon \mathrm{Spec}~\mathcal{O}_K\to X$ with the canonical metric given by $\|1_P\|_{\sigma}(Q)=G_\sigma(P,Q)$, where $1_P\in H^0(X_\sigma,\mathcal{O}_{X_\sigma}(P))$ is the canonical constant section. Further, for any vertical divisor $D$ of $X\to\mathrm{Spec}~\mathcal{O}_K$ we equip $\mathcal{O}_{X_\sigma}(D)$ with the trivial metric. By lineartiy, we obtain a canonical metric for any line bundle associated to a Weil divisor. If $D=D_{\mathrm{fin}}+\sum_{\sigma}r_\sigma F_\sigma$ is a Arakelov divisor, we denote by $\mathcal{O}_{X_\sigma}(D)$ the line bundle $\mathcal{O}_{X_\sigma}(D_{\mathrm{fin}})$ with the canonical metric multiplied by $e^{-r_\sigma}$.

Finally, the Arakelov intersection number is given by the weighted sum of the local intersection numbers at the several places of $K$, where the intersection numbers at the archimedean places are given by the Arakelov-Green function. Precisely, it holds $$(P,Q)=\sum_{v\in |\mathrm{Spec}~\mathcal{O}_K|} (P,Q)_v\log N(v) -\sum_{\sigma} \log G_{\sigma}(P,Q),$$ where $(P,Q)_v$ denotes the intersection number of the sections $P$ and $Q$ at the special fibre $X_v$.

This gives the intersection product for any two different sections of $X\to \mathrm{Spec}~\mathcal{O}_K$ and hence, by linearity, for any two horizontal divisors having disjoint support on the generic fibre. Any Arakelov divisor can be represented as the sum of a horizontal and a vertical Divisor, where vertical divisors are linear combinations of the components of the special fibres and the $F_\sigma$'s. The intersection with a vertical Divisor is easier to define and more or less what one would expect:

(a) $(D,E)=\sum_{v\in |\mathrm{Spec}~\mathcal{O}_K|} (D,E_\mathrm{fin})_v\log N(v)$ if $D$ is the linear combination of components of the special fibres and $E$ any Arakelov divisor.

(b) $(F_\sigma,E)=\deg_K E_\mathrm{fin}$, where $E$ is any Arakelov divisor and $\deg_K E_\mathrm{fin}$ is the degree of the restriction of $E_\mathrm{fin}$ to the generic fibre.

Here is a rather low-brow way of tracing through Arakelov's original ideas.

Recall that the intersection of two ordinary divisors $D,E$ can be written as $$ (D.E)_{v}=\sum^{r}_{i=1}-\log \lVert (f|E) \rVert_{p_{i}} $$ when $E$ is a horizontal irreducible divisor and $D$ is represented by $f$ on an open set $U$ containing the points in $\textrm{supp}(D)\cup \textrm{supp}(E)$ lying over $v$. (See Lang, page 73)

Arakelov's first original idea is that we can describe the intersection of two irreducible horizontal divisors at the archimeadean places by mimicing this construction. His starting point is that being irreducible horizontal divisors, $D, E$ are closure of points $P_1, P_2$ in the generic fiber of $X$. Then $P_1, P_2$ has residue fields $L_1, L_2$ with embeddings $\infty^{1}_{\alpha}, \infty^{2}_{\beta}$ that extends that archimeadean valuation $\sigma: K\rightarrow \mathbb{C}$. Further the number of distinct extensions corresponds to degree of $D,E$ over the generic fiber, and the image of $P_{i}$ under $\infty^{i}_{\alpha/\beta}$ corresponds to conjugate points in the Riemann surface $X_{\sigma}$ where $\infty^{i}_{\alpha/\beta}$ extends $\sigma$.

Arakelov's second insight is that because $P_1, P_2$ determines $D, E$ completely, the intersection of $D,E$ can be defined by defining the intersection of $P_{1}, P_{2}$. And this can be defined by defining the intersection of their images. Thus he came to the definition $$ (D, E)_{\infty}=\sum_{\alpha, \beta}(P_{\infty^1_{\alpha}}\cdot P_{\infty^2_\beta}) \label{VI} $$

By extending the first original idea, the natural thought is to look for functions that serves the local equation for point $P,Q$ respectively. And the function should have first order of vanishing on $P$ (or $Q$). Then the intersection index can be defined via $$ (P,Q)=-\log \phi_{P}(Q) $$ But this still does not fix the choice of functions available. The natural restriction is that $(P,Q)$ should be equal to $(Q,P)$. It is proposed by A. Parshin that one should look for functions which satisfies the Poisson equation $$ \frac{1}{2\pi}\Delta \log \phi_{P}dxdy=-d\mu $$ because then the difference of two functions, being harmonic, is a constant on $X_{\sigma}$. It is not too difficult to show that Green functions do have symmetric property if and only if we normalize their integral to be $0$ (See Lang, Chapter 2).

Therefore from this number-theory perspective we could have a lot of other choices as long as $(P,Q)=(Q,P)$, $\phi_P \ge 0$ and has a simple zero at $P$. The benefit of Green's functions is that it provides a fixed choice, but we may make other choices that suits our needs. If I am not confused, in laters works the admissibility condition is abandoned without much harm to the theory, for example. So there may be some room to explore in this area if you like!