How does chlorine in water evaporate?
Chlorine and Free Chlorine

Chlorine is the term commonly used to describe the addition of chlorine gas, sodium hypochlorite (i.e., bleach), or calcium hypochlorite (i.e., bleach powder) to water to form free chlorine. Free Chlorine is the sum of hypochlorous acid ($\ce{HOCl}$) and hypochlorite ($\ce{OCl-}$); a monoprotic acid with a $\ce{pK_a} = 7.54$. 

Chlorine gas ($\ce{Cl2}$) does not exist in water (except highly acidic) as it hydrolyzes to $\ce{HOCl}$. Chloride ($\ce{Cl-}$) forms from the redox reaction of $\ce{HOCl}$/$\ce{OCl-}$, but it is not reactive. 

Both $\ce{HOCl}$ and $\ce{OCl-}$ are volatile, with $\ce{HOCl}$ being $\approx\!5\!\times$ greater (Henry's solubility constant being $H = 0.076$ and $0.013$, respectively).$^{[\text{citation needed}]}$ When you open a pitcher of water containing free chlorine to the atmosphere, the free chlorine comes into equilibrium with the atmosphere and thus it volatilizes. As such, you do not lose "$100\%$ or all" of the free chlorine, but enough is removed that the concentrations drops below your taste threshold.



Side Note

If the water contains dissolved organic carbon and organic nitrogen (e.g., natural organic matter, pathogens), then the free chlorine could be reacting while it is sitting and thus you could be losing free chlorine this way as well. Further you can also add combined chlorine (e.g., monochloramine) to water in place of free chlorine. This also forms from free chlorine reaction with nitrogen compounds in water (e.g., urine) and is the common "pool smell" observed in swimming pools that are not properly chlorinated.