The Néron-Tate canonical height on elliptic curves

The local heights at nonarchimedean places are obtained as follows: you spread out the elliptic curve $E$ over the ring of integers $\mathcal O_{K_v}$, you spread out the two points into curves in this two-dimensional scheme, and then you compute their intersection.

The local height at an archimedean place is obtained by taking the complex points of the elliptic curve, and using a Green's function. However, you can fill out the elliptic curve to a solid torus (this is an analogue of spreading out over $\mathcal O_{K_v}$) and then realize your degree zero divisors (whose local height pairing you are going to compute) as boundaries of $1$-cycles in this solid torus. The Green's function should then have a geometric interpretation in terms of these $1$-cycles. (Sorry not to be more precise here; hopefully someone else can add more details. You can also look at Manin's paper Three dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry, where I think he proves a formula of the type I am suggesting. The rough idea, though, is that we can "spread out" at the infinite primes too, by passing from a torus to a solid torus, and replacing our points by $1$-cycles.)

Perhaps Chapter VI of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves" might provide some of the explanation that you're looking for. It appears to at least give some explanation regarding why local heights for non-Archimedean absolute values are constructed this way.