The bleaching action of chlorine occurs by:

Chlorine acts as a powerful bleaching agent in the presence of moisture. It reacts with water to release nascent oxygen ($[\text{O}]$):
$${\text{Cl}}_2+{\text{H}}_2\text{O}\to 2\text{HCl}+[\text{O}]$$

This nascent oxygen reacts with organic colored matter to break down chromophores through oxidation, rendering the matter colorless:

$$\text{Colored substance}+[\text{O}]\to \text{Colorless substance}$$

Because this process relies on oxidation, the bleaching effect of chlorine is permanent.