The survival of the smaller animals wasn't a subject of the programme, so this wasn't explored further.

But this got me to thinking... What environmental/climatic factors could have promoted the complete extinction of the larger air/land/sea animals but allowed smaller species to survive?

Last is reproduction, size is connected to reproduction, an elephant reproduces once every two years under ideal conditions, while a rat can breed every 24 days, so even if a few elephants do survive they repopulate so slowly they may never recover and some other minor event may finish them off.

So if you combine these not only are small animals more likely to survive they can recover far faster than most large animals, so all things being equal you should expect small animals to be better at surviving and diversifying after a event.