The textbook explains this in section 4.3.6 discussing trends in $M^{3+}$/$M^{2+}$ standard electrode potentials. The key concept is that half-filled $d^{5}$ configurations have extra stability due to exchange energy (explained in the context about Hund's rule and parallel spins). $Mn^{2+}$ already has this stable $d^{5}$ configuration, so it resists losing an electron. $Fe^{2+}$ ($d^{6}$) readily loses an electron to achieve the stable $d^{5}$ configuration as $Fe^{3+}$. The context explicitly states that 'the third ionisation enthalpy of Fe is lower than that of Mn' and that '$Mn^{2+}$($d^{5}$) is particularly stable' while '$Fe^{3+}$($d^{5}$)' has extra stability.