The answer to this question is actually quite simple. The departure of the men from the kitchen in the playTriflesmoves the plot forward because, when the men finally leave to investigate around Minnie Wright's home, Mrs. Hale and Mrs. Peters are left to their own devices.
This means that Mrs. Hale and Mrs. Peters were free to use their women instinct and inherent knowledge about the issues of women to identify themselves with Minnie Wright's situation. In fact, every time in the play when the men show up in the kitchen, they either interrupt, make a sarcastic comment, or degrade the conversation of the ladies.
SHERIFF: Well, can you beat the women! Held for murder and worryin' about her preserves.
COUNTY ATTORNEY: I guess before we're through she may have something more serious than preserves to worry about.
HALE: Well, women are used to worrying over trifles.
Conversely, every time they are gone, Mrs. Hale comes to a realization about Minnie which activates Mrs. Peter's own empathy. This is what makes the women make essential questions (instead of assumptions, like the men were doing) about what really went on inside that house, why, and how. This is exactly what happens. They actually find out everything they need to know. Instead of telling the men, the women form a moral bond and choose to keep it all to themselves. Minnie Wright may have committed a crime, but she had been put under such extreme suffering that she ended up snapping.