They flow outwards under their own weight into the ocean. In Antarctica, it is cold enough that melting at the surface is negligible. This means that almost all snow that falls ultimately must flow into the ocean. At some point at the edge, the ice loses contact with the ground and floats. This is the grounding line.
For a constant ice flow velocity, the amount of ice that gets spit out into the ocean is proportional to how thick the ice is at the grounding line.
Now, suppose that the slope of the bed gets deeper as you go from the grounding line into the interior of the ice sheet. Then, if the grounding line retreats, it's in deeper water. So, the ice flux increases. The increasing ice flux means that the ice sheet thins near the edge. Since it's thinner, it can't stay grounded in as deep of water. Imagine a big chunk of ice in a puddle of water that sits at the bottom of a puddle, versus a little piece that floats.
Now, since it can't float in as deep of water, the grounding line retreats because bouyancy is pulling up on the edge. This makes it go into deeper water, ice flux increases more, etc.
This could cause a collapse of the ice sheet, resulting in a few meters of rise. The timescale that this could happen is very poorly constrained (decades? centuries? a couple millenia?). It may have already irreversibly begun, or maybe not.
This is a typical cross-section[1]. To make it match what you're describing, I'd draw a slope from the grounding line and downward to the left, right? (Image depicts a flat bed underneath the ice, instead of a slope downward as you go inward)
[1] http://cdn.antarcticglaciers.org/wp-content/uploads/2014/09/...
EDIT: I think this image is a better representation:
http://cdn.antarcticglaciers.org/wp-content/uploads/2014/05/...
The terrain that this particular glacier is a bowl that is well below sea level. The more the glacier retreats the deeper the water gets (because the terrain is deeper), the more of the glacier can be floated, the faster the glacier can melt.
Imagine a pool that slopes from 3ft to 8ft full of ice melting. The thin part melts first, lifts up, water gets a little further underneath lifting the next bit of ice and melting it faster but because the pool gets deeper as you go the more ice can be lifted up and broken off so it melts faster as it goes.
Edit: yes, your second image is spot on.
Unless I'm mistaken.
EDIT: thanks, function_seven; pictures help the five-year old in all of us.
This is where you lost me. Bed? Flux? Can you explain what this paragraph means like I'm 5?
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?
Of course a likely collapse will be some fraction of that. Unfortunately there are big question marks about how quickly it could happen. But we do get very large icebergs every so often - for example a couple of years ago there was one the size of Delaware. (Of course floating ice to floating freely moving ice does not directly change sea level. It does make sea water a bit fresher though.)
(They call it Doomsday because they consider the worst scenario, when the inflow of warm water causes mass melting, and Thwaites and the ice held back by Thwaites leads to 10 feet of ocean rise. This 10 feet is also mentioned in the main article).
[1]: https://www.rollingstone.com/politics/politics-features/the-...
The worry is that in some cases melting glaciers may be released from these friction points causing an accelerated movement and rapid calving at the ice front. This will cause an even faster increase in sea level on top of the gradual thermal ocean expansion and freshwater runoff of meting ice that we are already experiencing.
An example of this is the Larson ice shelves on the Antarctic Peninsula that have had a series of collapses already. But luckily the Larson ice shelves do not have a huge sheet of ice behind them since the peninsula is rather narrow.
Thwaites Glacier has been identified as a potential "perfect storm" for global warming caused sea level rise. It has a wide (90-100 mile) ice front and and there is no narrowing or potential bottle-neck from side to side. It has an enormous ice mass behind it reaching to the center of West Antarctica. The main friction point currently holding it back is the grounding line where the ice meets the sea floor.
Between current grounding line and the center of West Antarctica there only seems to be one other ridge, known as the "Ghost Ridge" which is one of the subjects of the current expedition to determine if it is a true bedrock ridge or just silt and sediment deposited by the glacier. If the basin does in fact extend all the way back, the concern is that warm ocean water can work it's way under the ice and cause melting from below. As the ice melts it will float allow more water underneath. Meanwhile the ice front will calve and expose an even steeper front that will become unstable due to the height of the ice front. The icebergs will be washed out with the tide causing the process to continue.
Here is the most recent talk by Richard Alley I've found on the subject: https://www.youtube.com/watch?v=WE9Gqy8Yy9w