Some threadless headset designs have a compression ring to help keep everything together. There is a split in the ring or it has a bevel to it that is friction fit. If that compression ring does not come out first, the fork won't drop free without excessive force.

In the picture that you have provided, it is showing a cartridge bearing that is sitting in a race (the upper headset cup), the entire bearing should just lift out. You would then replace with a new cartridge bearing that fits your headset. Cannot really tell you more than that from the picture. Could you provide a make and model of your headset? Barring that if you provide and make and model of your bike, we can also look that information up.

The bearing system in your picture is a cartridge bearing setup, but the Amazon page that you have linked to shows a caged bearing headset set up. They are not interchangeable. Unless you want to replace the old headset for the new one from Amazon (shouldn't need to, unless the old one is damaged).