Diagnosis
The agitator dogs (also called agitator cams or splutch) are stripped out. These are plastic pieces inside the top of the agitator that engage with the drive shaft to make it rock back and forth. When they wear down, the shaft spins but the agitator doesn't move.
How to Fix It
Pull off the fabric softener dispenser from the top of the agitator. Underneath, you'll find a plastic cap. Pry it off to reveal a bolt (sometimes a special spline-head). Remove that bolt and you can lift the entire plastic agitator off the shaft. Inside the base of the agitator, you'll see the worn plastic dogs. The repair kit (part number like WPW10006355) includes new dogs and often the bolt/washer. Drop the new ones in, reassemble, and you're set.
If That Didn't Work
If the agitator now works but there's a loud clunking with each reversal, you might have installed the dogs incorrectly—they have a specific orientation. Take it apart and make sure they're seated in the tracks properly.
Heads Up
Don't overtighten the center bolt when you put the agitator back on. It's just supposed to hold the agitator down, not crush the plastic. Snug is fine.