How can I take the hooks off of this shelf holder?
Gently hammer up on the bottom, right next to the rail. They jam in place, especially if heavily loaded, and sometimes need persuasion to come loose again.

Several light taps will be less likely to go wrong than pounding as hard as you can.

Some (not all) will also respond to lifting up on the outermost end. Don't overdo that, or you'll bend them, especially if they are not of that flavor (which can be hard to tell for sure.)

They are called shelf brackets or shelf supports.