The formulae for the charge curve you showed above is [LATEX]V_c = 1 - e^{-t/\tau}[/LATEX] and the discharge curve is modelled by [LATEX]V_c = e^{-t/\tau}[/LATEX].
After one time constant, [LATEX]\tau[/LATEX], the charge curve [LATEX]V_c = 1 - e^{-t/\tau}; t \to \tau[/LATEX] becomes [LATEX]V_c = 1 - e^{-\tau/\tau} = 1 - e^{-1} = 1 - 0.3679 = 0.6321[/LATEX]
Likewise, After one time constant, [LATEX]\tau[/LATEX], the discharge curve [LATEX]V_c = e^{-t/\tau}; t \to \tau[/LATEX] becomes [LATEX]V_c = e^{-\tau/\tau} = e^{-1} = 0.3679[/LATEX]