The simplest way to explain it, is to simply say, you need to look where you want your horse to go. Just like any other discipline. The most effective way to get your horse to travel from point A to point B is to simply look at point B. It's amazing how effective a person's body becomes at getting a horse into position to travel that distance in the proper frame.
There are two styles of 'The Look' in barrel racing.
The first and probably the most recognized is the old classic style, where the rider runs to the barrel looking between their horse's ears at the rate spot on the pocket. When they get there, it's usually one more stride for the horse to be around the back side of the barrel(when they are all out running), then the rider turns their head and looks at the next barrel.
The second style of 'The Look' is more directed at keeping your eyes focused on specific points in front of the horse as he turns the barrel.
Either style is effective and correct, but it does depend on the horse you are running. If the style you use is not bringing the horse around the way he needs to be coming, then you need to look into doing something different.
I have seen more than one horse blow completely out of a barrel, the rider is looking at the next barrel and it is very difficult to get your horse back on track.
My problem however, is that I am very good about picking my rate spot, getting my horse there and then rather than using either of the appropriate looking methods, I am simply letting my eyes slide down my horse's neck and fixating them on the top of the barrel.
Similar to this...
Can you see how different this horse is turning the barrel vs. the horse in the first two pictures?
He's dropping his shoulder and bracing into the barrel. The other two horses are bending around the barrel.
It's a bad habit to get into and one that can be a bit difficult to get out of.
A horse will go where your eyes are telling them to go and with me looking down at the barrel, Moon, being the uber broke horse he is, is like 'You want me to roll over more? Okay!' and he is cutting back on top of the barrel. The ducking and diving between the barrels is happening because Moon is cutting to close to the barrel. I have become literally fixated on the top of that barrel and I'm still looking at it as Moon is trying to leave. He has no idea of where to go.
A more forgiving or solid horse may not react as much as Moon is to the problem, but this particular horse can go from 0 to 60 in a single stride and simply drop back to 0 in mid-stride. It's one of the things that makes him remarkable. But it does make him difficult, as it gives me little room for error.
Personally, I am more of a fan of the way Charmayne James and Ed & Martha Wright train themselves to look at specific points in front of the horse during the turn, rather than the sideways look, especially during slower than competition speed practice. But it does take more practice to get it right. Rather than try to re-explain it, here is an article written by Charmayne that shows her Axis Point pattern and her philosophy....The Eyes Have It.