How does oxidizing bleach work?

Because they are strong oxidizing agents, bleaches such as chlorine and sodium hypochlorite are able to break chemical bonds in many organic molecules.

If you have studied the interaction of light with atoms and molecules, you will know that different molecules and specifically the bonds in those molecules have the ability to absorb specific frequencies of light from the white light that is incident on the substance. When these colours are absorbed (and removed from the incident white light) the light that reflects back appears to have the complimentary colour to that which was absorbed.

For example, if a chemical were to absorb red light, the substance would appear green, since the light not absorbed (and hence, returning to the viewer) would now lack the red needed to produce white.

Therefore, the colour of a material is a function of the chemical bonds within it. The bleach can change the bond patterns within a material, making it no longer able to absorb the visible frequency that gave it the colour. It now appears white. Basically, the substance that produced the colour is still there; you just can't see its colour any longer.

It the same time, these oxidizing agents can alter the bonds in the material in such a way that the material is more subject to breaking down. This is commonly seen in bleaching stains from clothes, for example.

This same ability to break bonds within a molecule gives bleaches their ability to kill bacteria, and they find common use as antiseptics.