OK David, I love these challenges you throw at me. As a professional technical instructor I often find it necessary to explain difficult concepts without going into all the technical details or the math, even when teaching technical students. So without a treatise on naval architecture, or GM or GZ curves, here goes.
The example I'll use is small pontoon boat that was built to hold say 9 average people. We then will load the boat above its design capacity by first adding 3 more people, then 3 more again. But we will do this two different ways. The first way is to distribute the added load uniformly across the length of the boat. The second way is to add all the extra people to the bow section of the boat. The picture, yes I need to illustrate it, is shown below.
On the left side is the symmetrical loading. As more and more people are put into the boat, the boat will get deeper into the water displacing more and more water equal to the added weight. Despite the increase in weight and buoyancy, the center of all the buoyant forces (called the center of buoyancy, CB) will remain directly under the center of gravity (CG), the point at which all of the weight of the loaded boat appears to be concentrated at. (It's also the balancing point of the loaded boat.) As long as these two balancing forces remain acting on the same line one above the other, the boat will remain essentially stable.
On the right side of diagram we are loading the same number of extra people all forward of amidships into the bow area. This creates unsymmetrical loading and with it some problems in stability. By putting all the extra people forward, the added weight shifts the center of gravity forward. That means the boat not only has to displace more water a before, but it has to displace more water forward of amidships than aft of amidships in order to balance out the extra weight that was put there. Notice that there is more profile area below the waterline forward of amidships than aft of amidships. The boat will tip as shown until the new center of buoyancy falls directly under the new center of gravity. The total volume of water displaced by the boat on the right side of the diagram is exactly the same as the total volume displaced on the left side of the diagram because the total number of people in the boat is the same. However, the weights are shifted very differently as shown. With only 3 more people added, the boat on the right side is still stable but it is down by the head instead of being level with the waterline as in the case of the boat to the left.
Now suppose we add 3 more people forward. The center of gravity continues to move more forward away from amidships. The boat needs to settle deeper to displace more water to balance out the increased weight. But, as before, it also has to shift its center of buoyancy more forward to fall under the new center of gravity. But in this case an angle is reached where the under water profile area forward is not enough to shift the center of buoyancy far enough to fall under the new center of gravity. When this happens we have two forces acting in opposite directions (buoyance up, weight down) that just don't want to line up one under the other. The upward buoyancy force is to the left of the downward force of the loaded weight of the boat. The result, as shown, is a turning moment which causes the boat to rotate around what's called the center of floatation (CF) with the bow going down, the stern going up, until it achieves the vertical if the hull is strong enough not to break beforehand.
In the case of the Titanic, it was water flood into the ship that added the extra weight. Despite her head getting lower and lower as the night wore on, the ship was relatively stable until near the end when things started to develope a bit more rapidly.
A capsizing, by the way, is the same thing but a rotation in the transverse direction.
