Work done by a battery to move charge

Homework Statement

I'm working my way through MIT 8.02x, the intro E&M course, on EdX. In Homework 3/Problem 3, we insert a dielectric between plates in a capacitor that is connected to a battery supplying potential V. So the charge on the capacitor increases by Q. How much work is done by the battery to add that charge to the capacitor plates?

The Attempt at a Solution

I get the answer, but I'm having a conceptual problem. I understand how when charge Q moves across a potential difference V the work done is QV. But, in the situation above, how is the charge moving "across" a potential difference? The charges here remain inside conductors, and there can't be a potential difference inside conductors - except that of course there must be, because otherwise the charge wouldn't move from the battery to the capacitor. Can someone help me muddle through this?

I think my mind is still back in mechanics, where everything was force and motion. E&M is all about fields and potential energy, and I'm finding it difficult to shift my perspective.