Why must the hunters stay on the path and shoot selected animals?

The hunters are traveling far back in time to the dinosaur age for their safari. They have to be extremely cautious because if they in any way change even the tiniest detail of what happened in the past, they could change all of future history. As the tour guide, Travis, explains to Eckels:

The step of your foot, on one mouse, could start a series of events, the effects of which could shake our earth and future down through Time.

The safari guides have done careful research. They have chosen a T-Rex that they know was about to be killed by accident. Shooting it will not alter the future in any way. 
The path has been specially constructed so as not to touch any part of the terrain the time travelers pass through. As Travis explains to Eckels:

"And that," he said, "is the Path laid by Time Safari for your use. It doesn't touch so much as one piece of grass, flower, or tree. It's made of a special metal that floats six inches above the earth. This is to keep you from touching this world of the past in any way."

The risks the safari hunters are taking simply for the personal thrill of killing a dinosaur are very great indeed. As the story will show, the risk may have been too great.