What is the essence of such Wildean aphorisms (a witty saying that contains a general truth) as the following "[Women flirting with their own husbands] looks so bad. It is simply washing one's clean linen in public." and "All women become like their mothers. That is their tragedy. No man does. That's his __.

The lines quoted in question are taken from "Importance of Being Earnest"

The essence is that the writer is criticizing the action of people especially women for their behavior in public. He has done that in metaphorical way and quoting the practice of women doing flirt with husbands on public places during Victorian Era. He said that just like there is no need of washing a white cloth similarly women don't need to flirt with those, whom they already love