Preservation and bleaching of sliced mushrooms with hydrogen peroxide
Hydrogen peroxide is quite a strong oxidizer. First, let's address the bleaching effect: It bleaches by oxidizing colored compounds, making them colorless. From Wikipedia:


  An oxidizing bleach works by breaking the chemical bonds that make up the chromophore. This changes the molecule into a different substance that either does not contain a chromophore, or contains a chromophore that does not absorb visible light. This is the mechanism of bleaches based on chlorine.


This is also the mechanism of hydrogen peroxide as a bleach, since both chlorine-based bleach and hydrogen peroxide are oxidizers.

As for the antibacterial effect: hydrogen peroxide in this particular concentration (5%) may or may not be effective against bacteria (and viruses, fungi, etc.) due to the presence of the enzyme catalase in some bacteria and fungi. In those organisms which it is effective against, hydrogen peroxide would probably work by causing oxidative stress in the organism.