Here you go.
Thanks for asking this question. I really enjoyed the video clip.
It starts at the very beginning - explaining what factorial means.
Then if progresses to gamma functions.
so there is something for almost every one.
The is a clip from the Numberphile website I'll bet their other clips are just as interesting.
Here's a rather simplified "proof"
3! = 4! / 4
2! = 3!/ 3
1! = 2! / 2
0! = 1! / 1 = 1