The gait is called stotting or pronking. The legs stay relatively stiff, the animal leaves the ground with all four hooves and lands the same way, and each bound can clear a couple of metres of height and several of distance.
Three explanations compete and are not exclusive. Bouncing allows sharp changes of direction and lets the deer clear sagebrush, boulders and gullies that would slow a running animal, on exactly the broken terrain mule deer prefer. It raises the head, improving the view of a pursuing predator. And it may be a signal, an honest advertisement that the deer is fit enough to waste energy and therefore not worth chasing, an argument developed for gazelles by Zahavi and tested most closely in Thomson's gazelle.
White-tailed deer sharing the same range generally run rather than stot, which suggests terrain and predator type matter. Which function dominates in mule deer has not been settled.