This is an incredibly intense version of the push up. We are obviously getting more abdominal recruitment with this version than we would with the standard push up off the floor. A person will feel his or her core muscles contracting like crazy in order to stabilize the body while balancing through the arms on the ball. All of the body's segments must be integrated to function as a single unit in order to properly perform this exercise. That makes it very functional, as this is the way a person should be pushing something in the real world. If your back is arching, then you are not properly stabilizing your trunk or integrating it with the legs and upper body. Keep the torso rigid and pull the naval in for added spinal support.
This would certainly be a great exercise for certain athletes. It is however, very hard on the wrists. Great care has to be taken and it might take some time before a person can perform a push up like this on their toes with their hands on the ball. I've had many clients perform the much simpler version of this exercise, which is demonstrated in "The Training Room" page by a client. The ball is against the wall and the knees are on the floor, which makes a huge difference.