How to remove these brackets

It is entirely possible that this back bracket is glued on, unfortunately. Typically these have little pegs on the back that plug into holes in the back wall of the cabinet. Some installers decide to help keep them in place with adhesive, though that makes them much harder to repair later down the line.

If it is glued in then you could use rubbing alcohol to try to loosen it up some. That's usually my first go-to and it works more often than not. A local hardware store may also sell something specific to this application that you may find helpful.