Quite simply, yeast breakdown occurs when yeast cells are starved for nutrients (especially nitrogen). The yeast cells then devour each other to get at the nutrients they crave.
Just to clarify: While I agree with the actions John suggests, I don't think this is the actual mechanism for H2S production during the active fermentation.
First of all, yeast autolysis is real, but the key word fragment there is
auto-. The way it works is that, for whatever reason, a yeast cell dies. Then the
enzymes in the now-dead cell begin to break down that cell. It is not the case that starved, zombie yeast cells start eating the brains of other yeast cells. However, John is correct: if yeast autolysis (especially of a large amount of sediment) is allowed to proceed for a long time, then sulfur compounds may be liberated. But this is different from the H2S produced during an active fermentation.
From my reading of the primary and secondary literature, this is my understanding of H2S production and its relation to N deficiency. Proteins are made of amino acids, and two important amino acids contain sulfur. The yeast has to provide the sulfur to form these compounds to the proper organelle during protein synthesis. It does so in the form of H2S, which it extracts from more complex sulfur-containing compounds. One organelle passes the H2S off to the organelle responsible for protein synthesis.
However, nitrogen is a major component of amino acids (hence the root
amine, from
ammonia.) If there is a dearth of N, the organelle responsible for synthesizing the sulfur-containing amino acid cannot do its job; this results in a surfeit of H2S, which the yeast then excretes.
As John points out, we are extraordinarily sensitive to H2S and thiols (larger SH-containing molecules). That is why they add a tiny (ppm) amount of methane thiol to your natural gas supply, so that you can smell when you have a gas leak. This is good for gas-leak detection, but the unfortunate result for winemaking is that we cannot tolerate very much H2S excretion by our pet yeasts.
Hope this helps!