Tetrodotoxin blocks voltage-gated sodium channels, halting nerve and muscle signalling, and there is no antidote. It is produced by bacteria including Vibrio and Pseudoalteromonas species, which occur in sediments and in the invertebrates pufferfish eat. The fish accumulates the toxin and stores it, mainly in the liver, ovaries and skin, and is itself resistant because of mutations in its own sodium channels.
The evidence for the dietary origin is straightforward: pufferfish reared from eggs on toxin-free feed in isolated systems do not become toxic, and can be made toxic again by feeding them toxic material. Non-toxic farmed pufferfish have been raised commercially in Japan on this basis.
Inflation is a separate defence and is often misdescribed. The fish gulps water, not air, into a highly elastic stomach; air inflation happens only when a puffer is lifted out of the water, and trapped air can be difficult for it to expel afterwards. Puffers do not have functional stomach digestion in the ordinary sense, and the organ is specialised for this stretching.