Sadly, as the speaker, I would also like to know how that correspondence works. Long story short, some collaborators and I came up with a conjecture that certain pairs of symplectic varieties (algebraic varieties with an algebraic symplectic form) were in some kind of duality with each other. There is still not a good definition of this duality at the moment, but it seems to coincide with certain kinds of S-duality in physics. Unfortunately, you're going to have to wait a few months to read our paper on this stuff, and even then it will be a lot more conjecture than theorem, especially on this point.
So, how do we justify this conjecture? By certain remarkable coincidences. In the case of the quiver varieties and affine Grassmannians, these coincidences are manifestations of fact that both varieties are geometric avatars of the representations of semi-simple Lie algebras (read Nakajima for quiver varieties and Mirkovic-Vilonen for affine Grassmannians).
The most important bit of the conjecture is that these varieties are actually geometric avatars for the representations of categorifications of Lie algebras in the sense of Rouquier or Khovanov-Lauda, by which I mean that the simple module categories over these categorifications should occur as categories of modules over quantizations of either the quiver variety, or the slice. Actually many other module categories, like those corresponding to tensor products should show up as well. This is not written down anywhere in the literature, but the type A special case can be cobbled together from some different sources, and will be in a forthcoming paper of myself, Tom Braden, Tony Licata and Nick Proudfoot. Which I should now get back to writing.