The climactic moment when the two women find the bird with its neck wrung is when they both put all the clues together and realize that this was the reason Minnie Foster finally snapped and killed her abusive husband. Hence, they were looking at the motive of the crime which, if found by the Sheriff and the D.A., would be used as evidence that Minnie had a pre-meditated reason to kill her husband. As such, she would be tried as what we call today "Murder One," which is a crime punishable with death.
Mrs. Peters looked from the dead bird to the broken door of the cage. Again their eyes met. And just then there was a sound at the outside door. Mrs. Hale slipped the box under the quilt pieces in the basket, and sank into the chair before it. Mrs. Peters stood holding to the table.
It would put Minnie in a situation from which she would have no salvation. Nobody would have cared whether she had snapped or went momentarily insane. It would make her case even worse than it already was.
The problem is that in 1917, when this story was published, women's rights were just barely coming to be. A woman like Minnie would have found no mercy from the law, even if she were a victim of spousal abuse, which she clearly was. The bird would have been used against her to argue that her reasons to kill a man were not significant enough, or valid enough (animal rights were basically nonexistent then too) and thus she would be considered a butcher and a monster. An example of this would be the 1872 trial of Laura Fair, who killed her lover, and the way that her persona was trashed by the media simply because she was a woman who killed a man. Psychological spousal abuse were terms not even properly discussed in society. The media and society were not much different in 1872 than in 1917 either when it came to scandals of this nature.
In Mrs. Hale's own words, Minnie had changed for the worse after her marriage. The conditions of her home and her own personal mental and physical conditions evidenced deep psychological and physical abuse.
Therefore, the women hid the bird to do their part in saving Minnie however they could. But they also did it because they felt sympathy for her. They knew that the woman was living in chaos; they felt sorry and emotional about her situation, and knew that Minnie could have been any one of them.