I happily haven't seen these videos but it sounds like the defense posture they assume when hopping away is no longer an option. They aim their most toxic area, the parotid glands, towards the thing that's threatening to eat them. They may also hunker down, inflate, and remain immobile- probably in the hopes of not being seen and also being harder to swallow.

In other words, every time these people are doing this to their toad they're putting the fear of being eaten into it. I'm sure the toads don't find it in any way amusing.