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.