It's not exactly a new strategy, by any means. Age old Counter+Mime recipe. They've placed 8 pairs of those (all Mastered) on Barret, so when he gets attacked, he uses Mime 8 times. And since you can Mime your own Limit Breaks, guess what Barret does?
Barret has the Missing Score and Wizard Brace(let) on him. There's no Sneak Attack or Pre-emptive Strike happening here; the person doing it is just choosing W-Item as fast as possible, and dumping two Hero Drinks on Barret. Cloud *is* in Haste status though (Haste is what is replacing his Max HP, for the record, so he's likely wearing Sprint Shoes).
The Barret you see has 255 in all stats (definitely Str, Spr and Luck, at least) and L99. I also don't believe it's possible to get 9999 without using HP Plus *legitimately*, so that could be hacked too. But it doesn't matter; Str is all you need for this.
After two Hero Drinks, Barret has 408 Str and Spr. His base damage is thus 19338. Angermax does 1/2 of Base Damage (9669), and Emerald Weapon has 180 Def, so that drops the damage to 5877 to 6269 per hit of Angermax. Keep in mind that this can critical (quite likely to happen due to 255 Luck) to do 9999 damage in that hit. Assuming no criticals, it would take, at worst, 10 uses of Angermax to defeat Emerald. The more criticals you get, the less you have to use. As you saw, it only took 7 uses. The maximum number you could get out of this in one turn is 9 (8 Counters, plus a normal Mime command that you sneak in afterwards). Coupling that with the original use of Angermax, and there's your 10. Emerald is dead, whether he likes it or not.
And in case you're interested about what Emerald tried to do about all this... he tried Emerald Shoot, which would do 9520.875 damage max against a 0 Magic Def character. Except Barret has 408 MDf (255 Max, plus 2 Hero Drinks). Which reduces it to 1933 max. And Shell (from a Protect Ring, no doubt) reduces that to 906 to 966 damage.
Morale of this story: you can learn a lot just by seeing how much damage is done in a movie... and Counter + Mime has *always* been a sickening combination. Like I said, it's not a new strategy.
But, then again, keep this in mind. You need 8 mastered Mimes, and 8 mastered Counters. You need 255 Att. This is by no means simple to accumulate.