He can live comfortably on the inheritance his father left.
Literal
Father's left property [with-で] he [topic-は] comfortably can-continue-living.
父の残した財産 is a relative clause: 'the fortune that father left.' で marks means. 暮らしていける uses ~ていける, adding a nuance of sustainability — 'can continue living' — beyond the simple potential 暮らせる.