Learn English – What does the phrase “rushing off down” mean

Off of in that context means from.

So the children are copying from one another and the teachers are trying hard to stop it.

Off of is quite informal. It is being used here because it's meant to be a child talking or narrating.

It's a slightly quirky idiomatic usage that even many native speakers might not be familiar with.

What the writer means is that these "incidents" form the basis of interesting tales that can be recounted repeatedly in after-dinner conversations by a skilled raconteur (by implication, such as himself).

Because people like to be entertained in this way after a (formal) dinner, the host / organisers of the event would be likely to invite the writer. And there's usually the implication that any such "after-dinner speaker" would be likely to actually be paid to turn up (in addition to getting a free luxury meal, whereas the other diners are probably paying for the privilege of being there).

Sometimes there's no suggestion of payment / free food. I dined out on that story for years might simply mean When I went out for a meal with friends, I often used to tell that amusing anecdote, even if the speaker always scrupulously paid at least his fair share of the bill. He might just mean that his ability to entertain fellow-diners ensured such evenings were a success, and/or that people invited him out to dinner for this very reason.

Note that this idiomatic usage often includes other prepositions...

He dined off that for years
He dined out off that for years
He dined out off of that for years
He dined out on that for years
He dined out with that for years

All those (and probably more) seem fine to me, but others may feel different.