Has it been modified to give it more recoil?
Do more experience shooters handle this gun with more finesse?
Nitro Express 600 or 700, perhaps?
While a lightweight rifle in a big caliber will obviously have a lot of recoil, it also appears many of those shooter don’t know how to shoot a rifle.
If it has been modified, it probably has a lighter and shorter stock than normal. A combination of big gun, light stock, and the stock not fitting against the shoulder will bruise the hell out of the shooter.
Actually, I had thought it was an A-Square Hannibal rifle, chambered in .577 Tyrannosaur, fired at, IIRC, somewhere in the Persian Gulf area. Can’t remember if it was Bahrain, Dubai, or Kuwait.
Some ballistic data on the .577 T-Rex can be found here. I think they were the guys to first post the recoil videos too, but you need to be a member of their site in order to view them. The wiki for the T-Rex claims it has free recoil energy of 220 ft-lbf, FWIW. A lot of that is probably due to the rifle being light and muzzle-brake less.
Yep.
I’ve shot my .50 BMG rifle in the offhand position, and the recoil wasn’t that bad; I was able to easily control it. But that’s because it weighs 24 lbs. and sports a big 'ol muzzle brake. 
In all versions of this film that I’ve seen it’s been labeled as a .577 Tyrannosaur. As with all such big bore rifles, recoil is absolutely hellacious.
Here’s a guy firing a bolt-action and side-by-side .700 Nitro Express. This was an expensive video to make, given the around $60,000 cost of each of the rifles and the $80-per-shot price tag.
No, I’m not joking.
I wonder if they sell hand loading supplies for these big game rounds.
Me too. And those guys on the video don’t know to handle a gun properly.
Agree. But I’m also wondering if it may be due to the way they were taught to hold a rifle.
Generally speaking, for a right-handed shooter (i.e. trigger finger is right index finger), the left hand should not firmly grasp the front of the stock. The left hand is simply a pedestal to support the front part of the rifle. When I shoot, I don’t even grab around the stock with my left fingers; my left hand is flat like a pancake under the stock. This ensures I am not pulling the buttstock into my right shoulder with my left hand/arm. (For a right-handed shooter, only the *right *hand/arm should pull the buttstock into the right shoulder.)
This rifle has so much recoil, however, that it may require the shooter to firmly grasp the front of the stock with his/her left hand.
So what’s the intended purpose for such a gun? If you’re going to have to pick up the gun after every shot and probably not be able to move your right arm the next day, that doesn’t seem like a very useful weapon.
Is it sort of a macho “I can handle the recoil of this beast” thing?
Blue whale hunting. 
As far as the intended purpose, I imagine it’s a “because we can” sort of thing. Certainly that seems to be the purpose of making it available to shoot at that MidEast firing range.
I mean, the biggest critter on the planet commonly shot is the African Elephant, and that’s usually taken with cartridges like the .416 Rigby and .458 Winchester Magnum. (and professionals like K.D.M. Bell famously took immense quantities of various African game with cartridges such as the 7X57mm) I guess, all else remaining equal, a bigger hole is better, but I seriously doubt anyone is taking the .577 T-Rex hunting. (Unless it’s for Deathclaws…)
Some of these guys look like they have a little experience, just enough to get into trouble. I’ve known some folks that would have thought it the height of fun to tell someone that ‘it won’t kick much worse than a .22’