How does each of the following events advance the play’s action: the men’s departure from the kitchen, the discovery of the quilt pieces, the discovery of the dead bird?

The men's departure from the kitchen allows the women, the real "detectives", to gather real evidence of the marriage. The women see the clues of the abusive marriage, clues the men dismiss as "trifles". The dead bird represents the suffering of Minnie. The quilt: the act of knotting a quilt is like the act of killing a man with a rope around his neck.