One thing you didn't mention is the manifold in which this calculation happens: the Slodowy slice to a nilpotent of type $(n,n)$. This manifold is the key to everything, since it is a geometric avatar of the invariants inside the representation $(\mathbb{C}^{2})^{\otimes 2n}$.

In what sense is this true? First, we have an embedding of this tensor product into $\bigwedge{}^{\! 2n}(\mathbb{C}^{2}\otimes \mathbb C^{2n})$ as the elements of weight 0 under $\mathfrak{sl}(2n)$. By skew Howe duality, the invariants are the intersection of this weight space with the unique copy of the simple representation of $\mathfrak{sl}(2n)$ with highest weight $(2,\dots, 2, 0, \dots, 0)$. The Slodowy slice under discussion is actually the Nakajima quiver variety for this weight space, so that's one level of geometric avatarage; the Hamiltonian diffeomorphisms correspond to the action on invariants of the braiding for quantum groups, and the Lagrangians for crossingless matchings to the invariant vector for that matching.

This actually fits into a much more general picture: Lauda, Queffelec and Rose have proven that you always get Khovanov homology out of a category when you have a categorical action of $\mathfrak{sl}(2n)$ categorifying the simple with highest weight $(2,\dots, 2, 0, \dots, 0)$. Nakajima's construction of the action of $\mathfrak{sl}(2n)$ on the cohomology of the quiver varieties for this representation has an obvious candidate for a lift to the Fukaya category: he pushes and pulls on some Lagrangian correspondences, now just think of them as Lagrangian correspondences and look at the induced functor on the Fukaya category. There's a deformation quantization version of this that really works (http://front.math.ucdavis.edu/1208.5957), but the Fukaya category is a much spookier place. Seidel-Smith is in all likelihood just implementing the LQR construction there; of course, good luck to whoever wants to check that those correspondences really give a categorical action.

One thing you didn't mention is the manifold in which this calculation happens: the Slodowy slice to a nilpotent of type $(n,n)$. This manifold is the key to everything, since it is a geometric avatar of the invariants inside the representation $(\mathbb{C}^{2})^{\otimes 2n}$.

In what sense is this true? First, we have an embedding of this tensor product into $\bigwedge{}^{\! 2n}(\mathbb{C}^{2}\otimes \mathbb C^{2n})$ as the elements of weight 0 under $\mathfrak{sl}(2n)$. By skew Howe duality, the invariants are the intersection of this weight space with the unique copy of the simple representation of $\mathfrak{sl}(2n)$ with highest weight $(2,\dots, 2, 0, \dots, 0)$. The Slodowy slice under discussion is actually the Nakajima quiver variety for this weight space, so that's one level of geometric avatarage; the Hamiltonian diffeomorphisms correspond to the action on invariants of the braiding for quantum groups, and the Lagrangians for crossingless matchings to the invariant vector for that matching.

This actually fits into a much more general picture: Lauda, Queffelec and Rose have proven that you always get Khovanov homology out of a category when you have a categorical action of $\mathfrak{sl}(2n)$ categorifying the simple with highest weight $(2,\dots, 2, 0, \dots, 0)$. Nakajima's construction of the action of $\mathfrak{sl}(2n)$ on the cohomology of the quiver varieties for this representation has an obvious candidate for a lift to the Fukaya category: he pushes and pulls on some Lagrangian correspondences, now just think of them as Lagrangian correspondences and look at the induced functor on the Fukaya category. There's a deformation quantization version of this that really works (https://arxiv.org/abs/1208.5957), but the Fukaya category is a much spookier place. Seidel-Smith is in all likelihood just implementing the LQR construction there; of course, good luck to whoever wants to check that those correspondences really give a categorical action.