Remove any appliances or furniture in front of the backsplash and cover the floor with a drop cloth. While wearing a mask or respirator, apply adhesive remover with a paintbrush to the sides of tiles covering the damaged area.
Slide a putty knife behind the tiles and apply enough pressure that the tiles pop loose from the wall. If the tiles won't budge, use more adhesive remover. The adhesive remover softens the glue or grout holding the tiles in place, which allows you to scrape them off.
Sand the damaged spots of the wall with coarse-grit sandpaper. Rub a tack cloth over the area and sand with a fine-grit sandpaper. Clean off any dust that has attached to the wall with the tack cloth, leaving behind a smooth surface.
Hold a piece of adhesive-backed fiberglass mesh against the damaged area. Run a utility knife around the edges, fitting the mesh to the hole with 1 inch of extra space around all sides. Remove the backing, which exposes the adhesive side, and lay the mesh on top of the damaged spot.
Apply a thin layer of spackling compound onto the mesh with a drywall knife. After the compound dries overnight, add a second coat and dry again. Add additional coats, until the compound completely covers the mesh, letting each coat dry in between.
Rub sandpaper on the repaired area after the spackling compound dries. Keep sanding until the compound sits even with the wall. Wipe off any dust with a tack cloth and apply a thin layer of tile adhesive before attaching your tiles to the backsplash.