Origins
Some words barely change across millennia. Remain comes from Latin remanēre — 'to stay behind' — from re- ('back, behind') and manēre ('to stay'). The PIE root *men- meant 'to remain', and remain still means exactly that, four thousand years later.
The Latin manēre built a substantial English vocabulary around the idea of staying. A mansion is a place where one stays — from Latin mansiō, 'a dwelling, a staying-place'. A manor is the lord's permanent residence. Something permanent stays through all changes (per- 'through' + manēre). Something immanent stays within.
The legal word remainder, which sounds like dry legalese, is vividly physical in origin. A remainder estate is the portion that remains behind after a prior interest expires. In mathematics, the remainder is what stays after division. In bookshops, remainder copies are the ones that stayed on the shelf.
Later History
Remains — the noun — carries a particular gravity. Human remains are what stays after a person is gone. The remains of a meal, a civilisation, a fire. The word implies absence by naming what is present: remains exist only because something else has departed.
The word entered English through Old French in the 15th century. French itself eventually abandoned remaindre in favour of rester, but English kept the original, as English often does with French words that French itself discards.