This is a modern sporting metaphor, probably originating in the 20th century. It is a reference to the sport of tennis, in which you hit the ball over the net to put it in your opponent’s court. You can only hit or play a ball that is in your court. Therefore if “the ball’s in your court”, it is your turn to play and react. It is often used in business to mean that you need to make the next offer or in relationships to say that it is your turn to act.