Inexpeirenced fighters (I've been there) tend to just go crazy in a fight, hitting as fast and as hard as they can, throwing wild swings; just running off of pure madness.....
If taking a martial art though or even just having frequent sparring match with your freind in the back yard, you have one thing that can give you an edge against that insanity, Disipline.
To win at fights the main deal is to keep your head on and not go crazy like those lads there.
Like GroundAndPound2000 said, you've got to be able think and react fast. Thats where training helps.
If I where in a postion like that, where the was guy on top of me trying pound me to a pulp, my act would probably be to try and slide out from under him. People often leave "holes" in their pins (pin being a very loose connotation of the word judgeing from the video), weakness that are easy to exploit. Like the chap on the bottom there, had his legs up to push the attacker back, by twisting and moveing his legs one way and his upper body the opposite he probably could have rolled the guy over. Or at least put him in a less advantageous position. The Idea about pulling him close is a good one.