Why does Macbeth have Macduff's family murdered?
As Macbeth descends into insanity he becomes utterly obsessed with eliminating any threats to his power. In determining which threats to heed Macbeth becomes enthralled with all of the ideas that the witches give to him, believing that all of the their prophetic warnings will come true. It is clear that Macbeth seeks to eliminate threats to his own power due to the fact that he first murders Banquo and his children in order to ensure that Banquo's children will not pose a threat to Macbeth's rule (or that of his familial line). Moreover, in heeding the witches prophecy (who warn him of Macduff), Macbeth is required to take action against Macduff. When the murderers arrive to Lady Macduff's hiding place, they seek to find Macduff himself, perhaps not fully aware that he has fled, and instead find the family there. They kill the family in part to eliminate the threat to Macbeth, but also to send the message to Macduff that they will not fear him and that they believe him to be a traitor. Ironically, it is in this moment that Macbeth seals his own fate. By murdering Macduff's family he ensures that Macduff will retaliate and instigate an insurgency against him, ultimately concluding in Macbeth's downfall. This is the final act in the course of events that guarantees Macbeth's eventual death.
Macbeth has two reasons for having Macduff's wife and children murdered. The first is to enact revenge against Macduff, who has fled to England to join Duncan's son Malcolm and assist in raising an army to attack Scotland and place Malcolm on the throne. The second reason is to make an example of Macduff and discourage other men from deserting him. In Act 4. Scene 1, Macbeth says: "The castle of Macduff I will surprise, / Seize upon Fife, give to th' edge o' th' sword / His wife, his babes, and all unfortunate souls / That trace him in his line." This is typical behavior for a tyrant. It is the sort of terrorism Adolf Hitler was to employ during World War II when officers and enlisted men believed that the war was lost and were turning against him. Macbeth can no longer count on loyalty or patriotism and feels he must rule by fear.
Macbeth is sinking deeper into insanity. He is all consumed with doing whatever it takes to keep his crown. After he sees the witches again, he fears that Macduff poses the biggest threat to him. The witches warn Macbeth.
"Beware Macduff, beware the Thane of fire. The power of man, for none born of woman, shall harm Macbeth"
This frightens Macbeth so he wants to get rid of the threat. Macduff is in England helping Malcolm build an army. Malcolm is the son of Duncan, who Macbeth killed. Macbeth knows that Macduff's loyalties don't lie with him, and this makes him angry. He sends his murderers to kill Macduff's family. Macduff's wife is angry that her husband has left her and their children and tells her son that he is dead. A messenger shows up to try to warn her to take her children and leave, but she doesn't. She tells him:
"Whither should I fly? I have done no harm. But I remember now I am in this earthly world; where to do harm is often laudable, to do good sometime accounted dangerous folly: Why then, alas, do I put up that womanly defense, to say I have done no harm?"
Lady Macduff won't leave and in the end she and her children and servants are all killed. Macbeth thinks that, by the witches warning, Macduff was born by some supernatural force, however he was born by a cesarean section, so he was born by not supernatural deeds, but is the downfall of Macbeth after all.