This is bogus.
First off the black spot on Jupiter was a coincidence - not caused by Galileo. It was sent into Jupiter to avoid contaminating any of the moons (with possible life on them) with microbes from Earth... no other reason. There's no way that tiny amount of nuclear material could generate that big a blast under ANY conditions.
Jupiter is the only other planet in the solar system that is large enough to ever possibly start nuclear fusion (become a star). In fact scientists believe that it almost did, but couldn't get enough material and couldn't get hot enough for the process to start.
Saturn on the other hand, is too small to ever maintain fusion, and is way too light. Saturn is almost pure hydrogen, and is so light that it would float if you could find an ocean big enough to put it in.
If it was possible to make a gas giant planet a star simply by setting off a large explosion, then all the gas giants would be stars by now. Every time an asteroid or comet "hits" one of these planets, they leave scorch marks as big as the Earth (coincidentally probably what caused that spot on Jupiter). An asteroid/comet impact would create an explosion millions of times greater than anything 73 pounds of plutonium could cause. Planets get hit ALL THE TIME by asteroids and comets... Saturn and Jupiter especially because their huge gravitational pull sucks in more stuff than smaller planets (like Earth do). If this could happen it would have a long time ago.
And finally, even if this was even remotely possible, there is no way radiation from a star that is 3 times farther away from us than the sun and thousands of times less bright would harm us when the radiation from the Sun is already minimal. The Earth has a magnetic field around it that already protects us from the much more powerful radiation of the Sun... Even if this project existed and even if this somehow happened, there's no way it would directly kill us such as that video suggests. There might be some gravitational shifting or some different climate conditions, but not outright death by radiation...
And all this is not to mention why the hell would NASA do this? The probe was sent to study Saturn and specifically it's moon Titan... why would they even try to blow all that they just learned up?
Luckily enough we don't have to worry about any of this because it's impossible...
Go to your library and check out a child's level book on astronomy and you can prove everything about this theory wrong.
That turned out a lot longer than I was expecting, but I love astronomy and know oodles about it, so there you go!