Briefly, if it didn't lower the energy, it wouldn't happen. It describes a situation where the minimum energy is attained when the bonding electrons are not 100% on the anion and 0% on the cation, as in ideal ionic bonding, nor shared 50:50 as in a nonpolar covalent bond, but somewhere in between, e.g. 80:20. In fact there aren't really two completely distinct types of bonding, rather there is a continuum from covalent through polar covalent to ionic, with some bonds at or close to one extreme or the other, and some in between.
To look at it slightly differently, to say that the cation polarises the anion - i.e. that it exerts a force on the outer electrons of the anion, pulling electron density towards itself - is to say that the electrons can lower their energy by moving towards the cation. Remember F = -dE/dx.
Incidentally, just to be pedantic, lattice enthalpy is not exothermic, it is endothermic. It is defined as the enthalpy for the dissociation of the solid compound into gaseous ions - not the other way round. When you're writing a Born-Haber cycle, the enthalpy for the step ions(g)
solid is -L.