You have a sodium or lithium metal anode, and a sulfur-carbon "ink" cathode on a copper charge collector. The battery is encased in steel, so a cross section would be steel-Na|Li-glass-S (aq)-copper-S (aq)-glass-Na|Li-steel. The anode has to be sealed away from oxidizers and water, or it will react violently with a direct chemical reaction rather than through the desired electrochemistry.
During discharge, at the anode, the reactive metal violently throws its extra electron away as hard as it can, because it hates having that extra one. It really, really wants to have +1 charge. The steel casing doesn't help, but that glass electrolyte is apparently porous enough that +1 ions can pass into it. So the lithium/sodium throws an electron down the wire and jumps into the electrolyte, because positively charged ions repel each other. That works fine until the electrolyte fills up all those empty spaces with positive ions. The anode can't throw any more electrons down the wire, because positive ions have nowhere to go, and attract the electrons right back out of the wire just as strongly as the atoms could throw them.
On the cathode side, electrons are coming in from the wire, spreading out across a copper plate, and jumping onto sulfur atoms, which devour extra electrons with a passion. Sulfurs love to have 2 more electrons than they usually own. Sulfurs near the copper plate devour two free electrons and become [S]--. It just so happens they live in a thin "ink" cathode, and are therefore very close to the positive ions that have been filling up the electrolyte when they become charged. They get yanked across to the glass. Ordinarily, if all this happened in an aqueous electrode, each negative sulfur ion would coordinate with the two positive ions in solution so that the sulfur can keep its extra two electrons, and the lithium or sodium wouldn't have to take back their hated extra. The glass-aqueous interface probably prevents this?
I would guess that the glass electrolyte allows so much positive charge to be present in it, without allowing the sulfur inside, that as soon as the sulfur hits the glass, it fumbles its extra electrons that fly out of its grasp and home in on one of the positive ions, converting it back to neutral metal. Sulfur grumbles and goes back to the copper to pick up another pair. Repeat until the positively charged layer in the electrolyte is not strong enough to yank the electrons off of sulfur, and it hangs on to its electrons at the ink-glass interface. The battery is now fully discharged. Physically, you have layers Li, [Li]+ (glass), Li (glass), [S]-- (aq), Cu. I'm guessing here, as I haven't read the paper.
During recharge, you are pushing electrons into the anode and pulling them from the cathode. That reactive metal wants nothing to do with extra electrons, so it starts attracting positive ions back through the electrolyte toward the metal anode. Meanwhile, that copper plate is having electrons sucked out, getting positively charged, and yanking the sulfurs across to take their borrowed electrons back. The neutral sulfurs can then wander back to the glass and grab electrons from the reactive metal in it. They shuttle the electrons back and forth, turning the metal on the cathode side of the glass back into ions that transport back toward the anode. Eventually, the electrolyte is once again saturated with only positive ions, and the battery is fully charged. As soon as the charge is removed, the atoms in the anode will once again try to throw away their extra electrons and escape into the electrolyte.
It seems important that the glass electrolyte be constructed such that the cathodic atoms and ions cannot enter it. While the battery is charging or discharging, those will be bouncing back and forth like ping pong balls in a Chinese recreation center, shuttling electrons between the glass-ink interface and the ink-copper interface. The action of the battery is allowing the reactive species to ionize, and get their charges closer together, but not so close that they can coexist in the same ionic crystal or aqueous solution. That would make it too difficult to separate the charges again, to recharge.
Of course, as I am not a chemist, I might be completely wrong about this.