In the dissociation of NaOH into Na+(aq) and OH-(aq) ions, a process that absorbs energy, how is it that, in the end, heat is given off?

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What the specific details occurring in there?

First, breaking of bonds of NaOH:

NaOH (s) \rightarrow Na^+ (g) + OH^- (g)

If you think that it is strange to go to the gas state, think about the delta H lattice. It is formed from two gaseous ions which combine to form a solid ionic compound releasing heat. Well, this inverse lattice absorbs heat.

Then, there is heat of hydration. Here, the definition is the energy involved when one mole of a gaseous ions dissolve in a solvent. Since bonds are created between the ions and the water molecules here, heat is released.

And, heat of solution is equal to heat of inverse lattice added to heat of hydration;

\Delta H_{solution} = \Delta H_{hydration} + \Delta H_{inverse\ lattice}

Or, since the heat of lattice is the negative of heat of inverse lattice;

\Delta H_{solution} =\Delta H_{hydration} - \Delta H_{lattice}

I hope it helped! :)