Chlorine shows the bleaching action in the presence of moisture due to the formation of

Chlorine reacts with water to form HCl and HClO. But HClO (hypochlorous acid) is unstable and easily dissociates to form
nascent oxygen.
$\mathrm{HClO}\to \mathrm{HCl}+\left[\mathrm{O}\right]$

Nascent oxygen

Nascent oxygen is a more powerful oxidizing agent which is responsible for the bleaching nature.