Webster's The Duchess of Malfi is a pretty accurate reflection of the society in which it was written. It presents us with a rigidly patriarchal society in which women are under the firm control of men.
The values on which this society operates exist largely for the benefit of men and allow them to maintain control over women. If women should try to challenge these values in any way, then they are to be subjected to punishment, physical violence, and even death.
This is precisely what happens to the eponymous Duchess in the play. She has transgressed the norms and values of patriarchal society by appearing to have children out of wedlock. In actual fact, the Duchess's children aren't illegitimate at all; they were fathered by her husband Antonio.
But because the Duchess's wedding to Antonio took place in secret, the general assumption is that her children were born out of wedlock, which at that time was considered unacceptable, especially for a noblewoman like the Duchess.
The Duchess's brothers are so enraged by news of her apparently illegitimate offspring that they set out to restore the family honor by destroying their sister. When they hear about her secret marriage to Antonio, they're even more infuriated.
The Duchess has made her choice of husband entirely by herself, without the consent or input of her brothers, which would've been the done thing at the time. For someone of the Duchess's social position in those days, it would've been considered scandalous for a woman to choose her own husband, especially a social inferior like Antonio.
In the event, for all her undoubted bravery, the Duchess will pay for her transgressions with her life, a sign of just how firm a grip the patriarchy had upon society at this time.