Place the stool on its side so you can access the leg bottoms.
Wipe away dirt and dust from the bottom of the stool legs. Use a damp lint-free cloth.
Remove felt or plastic chair glides from the bottom of the stool legs. Felt chair glides can be peeled off by hand, whereas you'll need a flat-head screwdriver to remove plastic chair glides. Slide the head of the screwdriver between the stool leg and the plastic glide, which is like a giant tack. Twist the screwdriver to pry off the cap.
Hold the mounting plate flat against the bottom of the stool leg and use a screw gun to run screws through the plate and into the stool leg. Round mounting plates usually have a screw hole on the top and bottom, whereas square mounting plates will have a screw hole in each corner. Use this method to install the wheel mounting plates on all of the stool legs.
Install the wheels in the mounting plate. The base of each wheels will have a spoke with nubs near the tip. Run the spoke into the hole at the center of the mounting plate, and twist the wheel clockwise to lock the wheel into place. You'll feel the wheel lock into place when the nubs catch the locking mechanism.