To apply a simple animation to your org chart, go to the Animations tab.

With the chart selected, click the arrow next to the
Animate box to get the list of effects, which include
Fade,
Wipe, and
Fly In.

Select an option for making the org chart pieces appear on the slide. They can fade in all at one time, by each branch in the chart, by each level, or shape by shape.

As you point to an option, PowerPoint shows you a preview of the animation effect on the slide. The picture shown here captures a shape as it's fading in.
If you want to set up more complex animations, or change the speed of the one you've just applied, click Custom Animation on this tab. Also here are the transition effects for slides. Examples of these effects are a checkerboard pattern or wipe that occurs when you move from one slide to the next.