how one plays one's cards; card-playing strategy
Literal meaning: the way one plays cards, including strategy and decision-making in card games.
In poker, how you play your cards determines whether you win or lose.
how one plays one's hand; how one uses one's resources
Idiomatic extension: the way one uses available resources, options, or advantages in a situation, like playing a hand in a card game.
In negotiations, how you play your hand is crucial.
From カード (card) + の (possessive particle) + 切り方 (way of cutting/playing). The idiomatic sense is a metaphorical extension from card games.