You have what's called a 'universal' mount style monitor. First make sure all wiring (including the ground wire/strap) is unhooked from the monitor. If your cabinet has an access door/panel on the back, then open & remove that door/panel. Get a helper to give a push on the metal framework of the monitor. Your helper should just push the monitor out enough for you to get your fingers on the top & bottom mounting brackets. Be ready to tip the face of the picture tube towards you (it's heavy so be prepared). Never push/pull/strike the neck or neckboard of the picture tube.
Have an old blanket or large piece of foam rubber nearby to set the monitor face down on. That prevents scratching the face of the glass.