In this section, Kontsevich is describing a pattern of index contraction of a certain tensor which reproduces a statistical mechanics sum over configurations. For his model, you have at each vertex, which I will label by the integer k, a copy of the same four index tensor
$R_{a\alpha}^{b\beta}$
Where a and b are indices for one vector space of dimension $d_1$, $\alpha$ and $\beta$ indices are for another vector space of dimension $d_2$. The dimensions of the vector space are going to be the number of discrete states at each site.
You have two different directions, he calls them 1 and 2, but I will call them "up" and "left". The image in your head should be a finite square grid, and at each point there are two incoming arrows, one going up and one going left, and two outgoing arrows going up and left.
He multiplies together all the R's at all the vertices, and contracts the a-index of each site with the b-index of the site which is "left" of it, and the \alpha index of every site with the \beta index of the site which is "up" of it. This contracts all the indices in closed cycles, giving a complex number.
In the spirit of maximum generalization, he defines the construction on the most general graph that allows it: the graph needs up arrows and down arrows, and the up arrows make closed oriented cycles, and the left arrows make closed oriented cycles. Maximally generalizing in another direction, he notes that the same definition works for any objects with a tensor product and a notion of contraction, i.e. for tensor categories.
Sliding the vertices along the two kinds of arrows, you get two independent "flows" which are number conserving on a finite graph, so the trajectories of the flows are cycles, and on each cycle, the flow is a cyclic permutations of the vertices. For the Ising model on an square N-by-M lattice with periodic boundaries, one of the two flows just moves a horizontal line containing a site left by one unit, and the other moves a vertical line containing a site by one unit, so they commute as permutations of the $N^2$ elements, and he likes this condition, so he emphasizes it.
The partition function with this pattern of contraction is then
$Z = \Sum_{a_k,\alpha_k} \Prod_k R_{a_k}{\alpha_k}^{a_l(k)\alpha_u(k)}$
Where l(k) is the left-move on position k, and u(k) is the up-move on position k. That is, it's a sum over all possible values of the indices a_k, \alpha_k, of the product of a certain quantity which depends on the value of the indices at each site. In terms of the logarithm of R,
$R_a\alpha = \exp(-E_{a\alpha;b\beta})$, where E is the energy function, then
$Z = \Sum_{a_k,\alpha_k} exp(\sum_k E_{a_k\alpha_k;a_l(k)\alpha_u(k)})$
Where the big sum outside is over all the possible assignments of indices $a_k$ and $\alpha_k$ to each of the vertices, and l(k) is the left-map, while u(k) is the up-map.
To reproduce the Ising model, let $a_k$ and $alpha_k$ take the two "spin" values 0 or 1 (i.e. the two vector spaces $V_1$ and $V_2$ are both 2 dimensional). Then you want to make sure you get a zero contribution unless the index value $a_k$ is equal to $\alpha_k$. For this purpose, make the energy function $E(a,\alpha,b,\beta)$ infinite unless $a=\alpha$ (i.e. the corresponding R element is zero). Then the big sum on the outside collapses to a sum over $a_k$. The following values of $S$ are the finite ones:
$S_{00;00} = S(11;11) = 0$ (the up neighbor and the left neighbor are the same)$ $S_{00;01} = S(11;10) = J$ (the up neighbor is different)$ $S_{00;10} = S(11;01) = J$ (the left neighbor is different)$ $S_{00;11} = S(11;00) = 2J (both neighbors are different)$
The coupling J is the standard extra energy cost for mismatched neighboring spins.
Reconstructing what he is doing was a little difficult because he says "correspond" in a way that is vague on the top of page 22, and "an identification" on the bottom of page 21 without specifying the correspondence or the identification. Figuring those two terms out took me a little bit, and this is why I went into great detail.