Diagnosis
This is almost always because the little drive shaft on the handle has gotten worn down or broken off. It's the square plastic post that the brush head clips onto.
How to Fix It
Take the brush head off and look right at the center of the handle. You should see a square-ish plastic post. If it looks rounded off, cracked, or has a piece missing, that's your problem. Unfortunately, you can't fix the post itself. You have two options: 1) If it's under warranty, contact Oral-B. 2) If not, you need a whole new handle. The drive shaft is part of the motor assembly inside.
If That Didn't Work
Make absolutely sure you're using an official Oral-B FlossAction or CrossAction brush head. Some generic or other model heads don't seat properly on the Vitality post and won't spin right.
Heads Up
Don't force the brush head onto the post. If it doesn't click on easily, check for debris. Forcing it can snap the post.