Thank you for asking such a question that I would love to answer. I briefly describe the mechanism of H2S toxicity on page 133 in my book and cite a couple scientific references.
However, without getting into too much scientific mumbo jumbo... I can understand why H2S would be very toxic.
First, H2S like ammonia is an uncharged gas. Because of this one fundamental fact, H2S and NH3 can rapidly pass (without regulation) through cell membranes into the cell (In contrast, charged particles like NO3- and sulfate can't, because they have an electrical charge that doesn't allow them to penetrate the lipid barrier). The cell membrane is made up of lipids, which are fat-like substances that repel water and all electrically charged particles, that is, salts). Therefore, ALL cells (of plants, humans, fish, etc) have a tough time keeping H2S and NH3 gases from coming into their cells.
Second, H2S (hydrogen sulfide) rapidly and strongly reacts with metal ions to form metal sulfide compounds. Many cellular enzymes contain a metal like iron or zinc or copper as their "active center". The sulfide reacts with these metals to form iron sulfide, zinc sulfide, copper sulfide... and presto the enzyme is inactivated. Since enzymes are the workhorses of cells, the presence of H2S inside the cell would be devastating to cells.
Ammonia has a different mechanism. Ammonia gas getting into cells (again a neutral gas flooding the cell) inevitably causes the cellular pH to increase. The pH increase causes cellular enzymes, most of which like to work at physiological pH (7.2 to 7.4), to work much less effectively. At some point the cell can't function anymore and dies.
Thus, ammonia and H2S are toxic to all organisms.