This photo technique is called dragging the shutter. It intentionally blurs the image giving it a feel of motion. Panning the camera while dragging or slowing the shutter speed also has the benefit of blurring the background. This brings the focus of attention to the foreground – in this case the horse and rider. If you want to get real fancy, dragging the shutter and panning while blowing a flash with either front or rear curtain can really be a bunch of fun. HOWEVER, shooting a flash at a horse can cause a serious amount of trouble and a seriously pissed off rider if the horse is not “flash friendly” and comes totally unglued. In the right situation even the sight of a camera will freak a horse. A thoroughbred (with blinders) at the finish of and after a race is usually acclimated to a flash. It is part of their training. Besides the horse is usually so exhausted you could hit him with a brick and not have him freak.
Lastly. . . .punchy colors like RED help.