Cut or tear off a piece of the damaged armrest cover. Visit a piece goods store and find the upholstery section. Show them the old material and ask them if they have anything similar in texture and color. For my repair, I purchased a yard of marine grade vinyl, which is used to recover boat seats, for under $8.00. In retrospect, a third of a yard should be plenty of material unless you are prone to accidents.
Lift the armrest cover to expose the inside of the console storage compartment. On the hinge in the back, you will see three Phillips head screws. Remove all three and place them in a cup holder. You won't need these screws again until you re-install the armrest cover.
Very carefully pry these staples out of the plastic armrest base. If the plastic base is old, it may have become brittle and you don't want to crack it. When all staples are removed, throw them away and remove the outer vinyl cover. Inspect the underlying foam material. If it looks OK, don't worry about replacing it.
Flip the armrest right-side up and inspect your work so far. Does it feel tight? Are there any wrinkles? If there is anything that doesn't pass your inspection, remove the material and adjust it. After the next step, it will be harder and more aggravating to fix earlier mistakes.
Inspect the armrest cover once last time from above. The material should be stretched tightly around the entire perimeter of the armrest. The front corners should look round. Make sure there isn't any material sticking out that could interfere with the cover's opening and closing. If everything looks good, place the plastic panel back onto the underside of the cover and replace each Phillips screw.
Take the newly-covered armrest back to the Explorer and replace the three Phillips screws that you left in the cup holder back into the hinge. Open and close the armrest cover to make sure it has clearance on both sides.
How to Find the Door Security Code for a Ford Explorer