I'm afraid there's no push of a button. You can start out by designing the circular elements (in Illustrator for example)at the point where they fill the frame. Make sure they are centered to the frame. Bring them as layers in AE, and experiment with animating the size and rotation, the transfer modes between layers and with feathered masks that reveal the circular elements as they spin in.
This is obviously the short of it, it will take time and finessing to get the effect you desire.