The problem is that the damage is really close to the artwork. That and you'll have to paint the repair. Getting the color, texture, and sheen to match exactly could be difficult!
The big problem is going to be painting the repair so that it matches the rest of the cabinet. Getting the color, texture, and sheen to match exactly could be difficult! The artwork is very close to the damaged area, so my guess is that you won't be able to avoid damaging it during the re-painting process. You'll probably need to do some sanding and it will be very difficult to keep the sanding strokes from damaging the black and yellow area.
Anyway, the traditional way to repair something like this would be to cut the damaged area away in a rectangular shape. Then you get a new piece of wood, cut it to the size of the rectangle, and attach it to replaced the material you cut away. Typically the attachment would be done with glue (yellow wood glue or maybe epoxy) and some sort of mechanical fastener (dowels, biscuits, maybe pocket screws if you have access to the back. If there are small gaps between the repair piece and the original cabinet, they can be filled with Bondo. You then sand everything smooth, prime it, and paint it to match the rest of the cabinet.
FYI, you may find that the stickied thread on repairing water-damaged cabinets that appears at the top of this forum to be useful...