Your posting is a bit short on details but here is what you want to do.
Cut the 3/4 inch plywood so that it is at least the height of the cabinet and as wide as the space from the corner of the room over to the side of the existing other cabinets.
You will mount this rectangle of plywood flat against the wall with one side of it snug into the corner of the room wall. This plywood would be screwed through the drywall into the corner stud and the other stud. Use some good quality flat head wood screws of size #10x2.75", #10x3" or #12x2.75". Large screws like this will need a clearance hole drilled through the plywood. A pilot hole drilled into the studs will also permit easier installation and prevent the screws from splitting the studs. Four of these screws in each stud should be adequate. If you use a counter sink to bore a taper into the face side of plywood the flat head screws can draw in below the surface of the plywood. This will prevent them from interfering with the cabinet later.
Note that there is also access to the lower edge of the top plate inside the wall right along the corner with the ceiling. If you made the plywood tall enough to reach up to the ceiling you could also put some screws into the top plate too for more holding power.
Note that you should not place the plywood over any electrical boxes in such manner that the electrical box would become hidden. Another concern is that a laundry room is often near the the place where many electrical wires from the electrical panel fan out to other parts of the house in the walls and/or ceiling. For this reason do evaluate carefully so as to not be drilling or screwing into any electrical wires.
Once the plywood is in place you may want to finish it before mounting the cabinet. For raw wood make sure to use a good quality primer before applying the finish paint coats. You could also spackle over the countersunk screw heads so as to hide them and provide a nice flat appearance.
Mounting the cabinet will be a process of securing the back wall of the cabinet against the plywood. Most cabinets will already have holes in the back for just this purpose. If there are no holes then you will want to drill some. These holes should be the clearance size for the screws used to mount the cabinet. Appropriate screws (for a metal backed cabinet) are going to be round or pan head #12x1" wood screws. This size may penetrate the plywood and poke into the drywall some but that is OK. Slide the cabinet into place and mark the hole locations. Then drill pilot holes at those locations for the screws. Mount up the cabinet in place. I would recommend at least four screws if not six of them depending upon the size of the unit. Screws near the rear corners of the cabinet will be better than ones placed in the middle of the back. If there were existing holes that were too large for the heads of the screws you can select an appropriately sized flat washer to place under the screw heads.