The bleaching action of chlorine occurs via the process of:

Chlorine gas acts as a bleaching agent in the presence of moisture. It reacts with water to form unstable hypochlorous acid ($\text{HClO}$), which decomposes to release nascent oxygen ($[\text{O}]$):
$${\text{Cl}}_2+{\text{H}}_2\text{O}\to \text{HCl}+\text{HClO}$$

$$\text{HClO}\to \text{HCl}+[\text{O}]$$

This nascent oxygen oxidizes colored organic matter into colorless compounds, meaning the bleaching action is purely driven by oxidation.