First off, all Hydrogen Peroxide is not the same. Make sure you grab 3% Hydrogen Peroxide solution for this stain removal. Hydrogen Peroxide is a often used as a bleaching agent so keep in mind it might cause some discoloring. Thinner than water, hydrogen peroxide will penetrate the wood fibers helping clean deep stains and dry them as the solution evaporates.
When I tested hydrogen peroxide as a stain remover, I decided to use cotton balls to apply it to the surface. I dabbed a soaked cotton ball over the stain on the hardwood floor. I left this mixture for an hour at a time, reapplying 3-4 times to see a change.