The comments above rightly iterate the fact that snakes want to be left alone and they would use whatever natural mechanism to evade. Coming to your questions:
Is the man telling the truth, that the thirty thousand fatalities per year, were all from trying to grab a puff adder with their bare hands?
This can't be true. Puff Adder snakes have the tendency to rely on their camouflage rather than slithering away. However, it is just unsafe to assume that a snake won't bite back despite stepping on it. To be on the safer and sane side, don't do it. I'd assume that the guy in the video doesn't seem to have ill intentions behind providing this slightly misleading information. The only rationale I can think is, to educate the people with the fact that snakes are dangerous but aren't necessarily enemies of mankind. I'd start with saying "If you leave them alone, they mean no harm to you." - may be the guy intends to say the same in different words.
What's going on? Was that particular snake just trained to be tame?
It isn't enough to set a benchmark for all the Puff Adders. The behavior would change with the same species of snake and from situation to situation. I have observed dangerous snake such as Saw-scaled viper (which is one of the most ferocious biting snakes) exhibit different behavior from time to time. When it was captured, all it wanted to do is strike. Then a couple of days in captivity meant it was agitated already. It was fed and then for some reason it was moved. And we saw that it had regurgitated the lizard and was only looking to get away without striking back. So, you can't have such a small sample size for an experiment and conclude. I am certain that multiple such experiments would yield astonishingly different results.
Is it really representative of what they will typically do?
Not really, you can't rely on any such observation. Its just safe to assume that best possible thing to do is to leave them alone.
A bit of reading about the snake itself makes a few facts clear.
Quoting from Wiki:
The scientific name of Puff Adder is Bitis arietans. The word arietans means "striking violently" and is derived from the Latin arieto.