How can I fix these peeling cabinets?
You are going to have to re-finish. If it is just the doors, I have seen people leave the cabinet face-frames alone and just refinish the doors. If it really is particle board there will likely be no way to restore them to an acceptable "wood-look" finish; you will probably need to consider painting them.

To paint, make sure you sand all surfaces until any material that is peeling, AND any material that looks like it might peel, is removed. Fine sand to knock down any gloss, then wipe the surfaces clean with mineral spirits (paint thinner), turpentine, lacquer thinner, or some other solvent. Prime with a high-hiding latex binding primer like BIN 1-2-3, then use a good quality gloss or semi-gloss paint with a professional grade paint brush (don't skimp on the paintbrush for the finish coat, your end result will be way better).

it's a lot of work, but nothing good is ever easy.