As night drew near, people began to head for home.
Literal
Night [subj-が] approach-[and-て], people [topic-は] path-home [to-に] set-out-begin-[past-た].
家路につく ('take the road home, head home') is a literary set phrase — 家路 is the home-road, つく is 'reach, set out for.' Combined with ~始める for 'begin to.' 近づく ('approach, draw near') is itself a compound from 近い+つく.