Salt based chlorine generators for residential swimming pools use
electrolysis to breakdown NaCl into sodium and chlorine. Please
explain how this process works and how does the base elements of
sodium and chlorine recombine to form salt again?

Sadium hydroxide forms at one electrode and chlorine gas at the the
other if direct current is used. A semipermiable membrain keeps the
chemicals mostly separate. More commonly the membrain is ommited and
ac is applied to the electrodes so mixing occurs and sodium hypro
chlorite (bleach) is produced at both electrodes. This is the pool
sanitiser. It releases one atom of very active oxygen which distroys
many kinds of bad stuff in pools and drinking water. Some times
complex chlorine compounds called chloramines are formed, but this is
undesirable. When sodium hyprochlorite gives up it's oxygen atom, it
becomes sodium choride which adds to the TDS = total disolved solids
in the pool. It is rare that TDS become excessive unless the pool
water supply has a high mineral content.   Neil