With the quick caveat that I’m not a doctor; from what I remember, most paralytics will take effect in under a minute, and kill the victim in under five. Most of these will cause respiratory arrest. You’ve paralyzed them, so they can’t go anywhere, but you’ve also paralyzed their lungs, so they’ll also stop doing that pesky breathing thing.
If you’ve got someone on life support, there are some real medical applications for this. Particularly in surgery. When it’s administered in the field, they’re dead.
You’ll also find a few animals that administer paralytic poisons. This is some seriously scary stuff. Same problem though, in higher forms of animal life (read anything with lungs) it will stop respiration and result in the victim asphyxiating because their lungs are paralyzed.
The mode of action, as I recall, is that the poison actually interferes (or blocks) the neurotransmitters responsible for muscle control. It doesn’t matter how badly the victim wants to move, their body can’t get the message.