Why is the formation of NaCl an exothermic process?
The negative standard enthalpy of formation of sodium chloride (i.e. ‘the formation is exothermic’) is primarily due to the formation of a crystal lattice. This crystal lattice is a macroscopic arrangement of ions in a cubic fashion with each sodium ion surrounded by six chlorides and each chloride ion surrounded by six sodium ions.

If it were not for the generation of this lattice and its inherent stabilising properties (a very good charge distribution, charges neatly alternate, contact between species of identical charge is strongly inhibited) the formation of sodium chloride would be much less exothermic.

Note most importantly, that it is not merely the generation of sodium cations (that process is endothermic) or the generation of chloride anions (that process is only weakly exothermic) that causes the overall exothermicity of the entire reaction.