The bed is the ground below the glacier. In this case, it is below the sea level. The glacier is thick, so there are parts where it is not floating, but actually lying on the sea floor. (You can probably imagine what happens if you put a lot of ice in a shallow pool.)
Because the glacier is thick (non-Newtonian fluid btw! this means it will change it's 'liquidity' if you put stress on it.) it will flow outward. At some point it will be thin enough and/or the sea floor deep enough that it can float on the sea water. The place where that happens is important, because floating ice displaces water, so it will not raise the sea level anymore if it melts (which is why we're not too worried about the Arctic ice melting, at least with concern to sea level rise). This place, or rather line since it is a 3D glacier, is the grounding line.
In general, flux is the amount of something flowing through a set point or line. (e.g. If you stand next to a river, the flux is the total amount of water passing you in, say, a minute.) In this case, we watch the flow of ice through the grounding line. The velocity of it should be constant, so the same speed always, because the ice is flowing from the center and is pushing constantly.
This means that if the ice is thicker there, you get more flow because a higher amount of ice can pass at the same time. (Imagine a river with the same water speed but twice the depth - more water passing by.) The ice flowing into the sea breaks of or melts and is thus lost to the glacier. So if the grounding line is in deeper water, there will be more ice mass lost over the same time period.
Normally you'd expect the sea to get deeper the farther you get away from the glacier. In some cases, and quite often in West Antarctica, this is actually not the case. The further you get 'inland', the deeper the sea gets. (Still covered fully in ice that is lying on the sea floor/bed.)
This is an unstable situation. When more ice is lost at the front due to melting etc, and the place where the glacier still lies on the sea floor (grounding line) retreats a little, at this place the glacier will now be thicker than before (same height at the top + lower ground = 'deeper river').
The ice speed stays the same, so now we lose more ice at the grounding line. This makes the ice thinner at the edge (because more ice is lost than before). If it gets thinner it can float more easily, and the grounding line goes back a little. This is a run-away process until we reach an area where the sea does not get deeper anymore.
Does this help?