It's a combination of stirrup use, center of gravity, extreme muscle control, and a very close working relationship with the horse. In this video, there's a very high likelihood that the archer trained to make this shot on the same horse that they're riding for the final take, and that the horse is already highly trained to carry and recognize archers.
In order to fire an arrow from the back of the horse accurately, you basically have to be so experienced with horseback riding that you can ride a horse running at any speed without ever allowing your butt to touch the saddle, or for the horse's gait to jostle you above waist-height. It's very possible to do, but it's incredibly difficult to learn to do well and maintain that position/pivot to adjust your aim over longer than a few minutes of riding and shooting while also doing everything you can to avoid bouncing as the horse moves. The best archers out there can accurately fire an arrow while standing on their horse's back as the horse is running at a full gallop.
The stirrups are there to help a rider correct their center of gravity if it goes too far in one direction, as well as to signal the horse to turn, speed up, or slow down with specific signals given through the rider squeezing the horse's body with their legs, tapping them with a riding crop (riding crops should NEVER be used hard enough on a horse to hurt them; just to lightly tap the horse on one flank or the other), and/or combination signals through the horse's reins and the pressure they feel from the rider's heels in the stirrups.
The really interesting part lies in the fact that the archer must trust the horse they're riding in order to successfully ride, shoot, and not fall off. I never got into archery when I was riding horses, but I was a barrel racer (where you ride a horse as fast as possible around 3 triangularly arranged barrels so fast that the only thing that keeps you attached to the horse is centripedal force and Death blinking long enough not to see what you're doing). A horse who knows its rider, or is just extremely experienced with carrying people, can modify its own gait and center of balance to help protect their rider from falling.
Horses know when you don't know what you're doing if you're going on a trail ride you payed to go on in a group. They can tell just by feeling you in the saddle whether or not you know how to ride, and if they can trust you to not screw up, fall, and die. The horse is 100% aware that "if the human falls off, they'll be hurt or die", and if the horse likes you, it will try its best to compensate for mistakes you make while riding.
In the above video, the archer is only using one stirrup to support their full weight on the horse due to the angle of the shot. The video is of a classic "drive by" arrow shot, such as if the archer were riding past another enemy archer or mounted warrior in combat. Their right leg (the foot that's actually visible in the shot) and foot are in a prepared position so they can quickly correct their stance and balance themselves again after they've fired the arrow into the target. Only one stirrup is keeping the rider on the horse, and if you watch their left leg carefully, you can see their knee subtly bending in response to every time one of the horse's hooves is about to strike the snow. This effectively neutralizes the jolting energy from the horse throwing off the archer's aim (and the archer themselves), and keeps the rider steady and stable while they aim and fire the bow.
If you look even more carefully, you'll also notice the horse responding to the archer's movement: The moment the archer releases the string to fire the arrow, the horse stops running so aggressively (nostril breathing evens out, front legs stop rising as far into the air with each step). This is to slow down the forward force of the charge and provide an even more stable gait so the archer can safely sit back in the saddle without the bouncing of the horse's galloping jostling the still-unstable archer as they're trying to stabilize their center of gravity in the saddle and take on a more comfortable posture. There's a good chance the horse already knew it was time to slow down from the sound of the "twang" of the bowstring and began slowing down the second it heard it!
Writers: When you're writing characters that work closely with horses, it's absolutely critical to emphasize the relationship between the rider and their horse(s), and to be aware that, with horseback riding, the horse itself is its own character.
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