means A close friend or companion (especially in British and Australian usage), or one of a breeding pair, or the act of breeding — and, separately, the chess-shortened term for checkmate, the move that wins the game.

from Two threads tangle here. The 'friend' sense comes from Middle Low German 'mate' or 'gemate,' meaning a messmate — literally one who shares your food, from the same root family as 'meat' (which once meant food in general). The chess sense is a completely different lineage: it's clipped from 'checkmate,' which travels back through Old French 'eschec mat' to Arabic 'shah mat,' itself borrowed from Persian — and the popular gloss is 'the king is dead,' though scholars note 'mat' more accurately means the king is helpless or defeated rather than slain. So your dinner companion and the killing blow only became homophones by accident of history.