Determine which prong of the upholstery staple remover fits best with the staple you are trying to remove. It should wedge beneath the staple easily. Some staples will give more easily when you use the sharp prong, and some will do better with the dull prong.
Wedge the staple-remover prong beneath the staple and press down to create leverage and lift the staple from the chair.
Pull straight up to remove the staple. Do not wriggle the staple remover, as this may damage the wood. If you need to, use a pair of needle-nose pliers to finish removing the staple.