The stock design and shooting position are factors as is the firing position and if there is a muzzle brake or not.
A stock with a low butt pad will have more muzzle jump than a stock with a high butt pad, kind of like an AR 15 or Cadex.
Another factor is scope magnification. If at a low zoom level you will have a wider field of view, so even after recoil you can still easily see the animal.
If the rifle is well controlled, not only can you see the animal get hit, you can see the bullet in the air during flight. You are not really seeing the bullet itself, you can see humidity condense momentarily by the compression wave in front of the bullet. Having said that, this is much easier to do at longer ranges when the bullet spends more time in the air, than at only 100-200 yards.
As for seeing the hit before recoil, that's BS. He's probably just got good follow through and just stays focused on his point of aim throughout the recoil. There is no shooter skill that will overcome the realities of physics.