Pour a small amount of wood glue on a piece of scrap wood or in a small disposable container.
Lift the splinter up carefully with the tip of a blade or a pair of tweezers. Hold the splinter in place, but be careful not to pull it off of the cabinet.
Dip the end of a toothpick in the wood glue. Spread the glue in the hole made by the splinter.
Push the splinter back down against the cabinet. Hold the splinter in place while you wipe away any excess glue that comes out as the splinter refills the hole. Place a flat piece of scrap wood over the splintered area and clamp it down with a wood clamp.
Allow the glue to dry completely. Some wood glues dry faster than others, so check the label for information specific to your product. Remove the clamp.
Inspect the repaired area. If the splinter is perfectly flush with the rest of the wood and you can't tell the cabinet was damaged, leave it alone. Otherwise, sand the area with fine-grade sandpaper until it is perfectly smooth. Wipe the wood with a tack cloth and apply wood finish, such as varnish. Use the same product that was originally used to stain the cabinet.