What's so special about the simple organophosphate group that makes them able to do it?
Acetylcholine's normal function is to cleave acetylcholine's ester bond to convert it into acetic acid and choline, but certain compounds can interfere. The short answer is that these nerve agents all share similar structural components allowing them to be bound by acetylcholinesterase, where they then either react to alter the enzyme's binding site, preventing activity, or are just very slow to hydrolyze, temporarily inhibiting the enzyme until it can remove the compound.
Not all of these related acetylcholinesterase inhibitors work in exactly the same way, but if you examine their structures and compare them with acetylcholine, the normal substrate of the enzyme, you'll notice some similarities:
You can see that they all have this central double-bonded oxygen (a few have sulfur instead, but are converted to oxygen metabolically), which is important for fitting into the enzyme. The ester bond holding the choline to the carbonyl in acetylcholine is relatively easily hydrolized, but phosphoesters are much slower to hydrolyze. From what I can tell, this seems to be one mechanism behind trecresyl phosphate's toxicity and probably why it's not as acutely toxic as something like sarin or VX. Sarin, on the other hand, has a pretty good fluoride leaving group. This lets it form a strong covalent bond with a serine's sidechain in ACE's binding pocket. This reaction is effectively irreversible, making sarin very dangerous as it totally disables any ACE it reacts with, meaning it takes very little to cause a significant effect.
Acetylcholin’s cleavage proceeds via the attack of a nucleophile (possibly enzyme-activated water or a nucleophilic side-chain of the enzyme) to the carbonyl function resulting in a tetrahedral intermediate with a negative charge on the former carbonyl oxygen.
The phosphate groups in various inhibiting compounds are very similar to this tetrahedric intermediate because:
They have a rather positively charged atom in the middle (phosphorus) like carbon would be in acetylcholin;
they have an oxygen bonded to said central atom with a formal negative charge;
they have at least one further oxygen atom that is able to accept hydrogen bonds;
the overall structure is tetrahedral; and
the bond lengths and thus the shape of the phosphate group are rather similar to those of the tetrahedral intermediate of acetylcholin.
Thus, these phosphates are transition state mimicking competitive inhibitors. Enzymes usually happen to stabilise the transition state well (it is one of the most efficient catalytic mechanisms) so anything that mimicks the transition state will usually bind well in the active pocket. However, since the phosphates’ structure is stable, it cannot react like acetylcholin could; thus the transition state cannot break down and it is hard to liberate the inhibitor. The enzyme is almost permanently blocked.