From Old Norse 'flatr' (level), from PIE *pleth₂- (to spread out) — apartment sense from 'floor of a house.'
From Old Norse 'flatr' (flat, level), from Proto-Germanic *flataz, from PIE *pleth₂- (to spread out). The apartment sense from Scottish 'flet' (floor of a house). Key roots: *pleth₂- (PIE: "to spread out").
A 'flat' (apartment) comes from the Scots word for 'floor' — each floor was a separate dwelling.