Ultimately, she and her son exit, although through different means: "[Stage directions]. Exit LADY MACDUFF, crying 'Murder!'"

The previous scene to this one, Act IV, scene i, ends with Macbeth swearing to act according to his heart's purpose as soon as he thinks a thought. The thought to hand for Macbeth now is that all living persons in Macduff castle should be killed so no lineage of Macduff's should carry on from that day forward.

MACBETH. To crown my thoughts with acts, be it thought and done:
The castle of Macduff I will surprise;
Seize upon Fife; give to the edge o' the sword
His wife, his babes, and all unfortunate souls
That trace him in his line. No boasting like a fool;
This deed I'll do before this purpose cool.

To examine the foregoing in some detail, in the opening of IV.ii, Lady Macduff and her young son are listening to and she is speaking with Ross, Macbeth's cousin and a Scottish nobleman who chooses to reject Macbeth and join Malcolm, Macduff and the English. While Macduff's son is listening, Lady Macduff and Ross are debating whether or not Macduff's flight from home was traitorous. Lady Macduff says that he acted like a traitor: "What had he done, to make him fly the land? [...] Our fears do make us traitors." Ross rebukes her, replying that it was as likely his wisdom that made him fly: "You know not / Whether it was his wisdom or his fear." Lady Macduff challenges Ross, saying that wisdom doesn't abandon love, although fear leads to flight--and flight manifests only fear--and that flight that betrays reason shows no wisdom.

LADY MACDUFF. Wisdom! to leave his wife, to leave his babes,
...
All is the fear and nothing is the love;
As little is the wisdom, where the flight
So runs against all reason.

After Lady Macbeth laments that Macduff is dead (as she assumes must be the case and as she metaphorically asserts to be the case: he is a traitor to lie about his love then to betray her, and traitors die) and that his son is fatherless, Ross, asserting that it would be his "disgrace" to stay (since she is asserting what Ross believes to be untruths about Macduff), quickly takes his leave. She and her son debate his fatherless state, ending with the son's claim that Macduff is not dead (thus not a traitor by her own argument): "If he were dead, you'ld weep for him: if you would not, it were a good sign."

Upon this note, the Messenger, a person unknown to Lady Macduff, walks in and apologetically delivers a warning message to her, advising her to take her children and leave. The use of the word "doubt" indicates that he feels unsure about whether danger approaches or not. His suggestion of advice, "If you will take a homely man's advice," underscores his feeling of uncertainty: if he had been certain that Macbeth could have fallen so low as to order the deaths of lady and children, his "advice" would have been more in the form of a command, whether he was known to her or not. Still demurring, he regrets being "savage" by causing her fright, then begs for her preservation and hastily departs. His actions speak louder than his demurring because, in the end, he "dare abide no longer," giving the proof to his encroaching conviction that Macbeth could indeed have been bent so far as to murder a whole family.

Messenger. Bless you, fair dame! I am not to you known,
    Though in your state of honour I am perfect.
    I doubt some danger does approach you nearly:
    If you will take a homely man's advice,
    Be not found here; hence, with your little ones.
    To fright you thus, methinks, I am too savage;
    To do worse to you were fell cruelty,
    Which is too nigh your person. Heaven preserve you!
    I dare abide no longer.
Exit

Astounded and bewildered, she puzzles aloud over where she has to go to and why she has to go, concluding that to do "no harm" is inadequate when "to do harm / Is often laudable" and to "do good sometime [is] / ... dangerous folly." It is at this moment, while Lady Macduff is caught in the grip of a cruel conundrum without answer or solution, that the murderers walk in looking for Macduff (but under Macbeth's orders to kill "His wife, his babes, and all unfortunate souls" who abide in Macduff castle), asserting that he is a traitor (which Lady Macduff asserted earlier to Ross, although--when compared to the murderers--clearly in a more philosophical or metaphorical vein). Young Macduff, after having loosened his tongue up through witty exchange with his mother, shouts out in his father's defense: "Thou liest, thou shag-hair'd villain!" His reward--and the last thing that happens to him in this scene--is to be slain by stabbing and accused of "treachery": "[Stabbing him] Young fry of treachery!" In a bitter death scene, he has time before he dies to cry out to his mother, "He has kill'd me, mother: / Run away, I pray you!" The last thing that happens to Lady Macduff is that she runs off "crying 'Murder!'" with the murderers in deadly earnest pursuit after her.

It is important to note that Lady Macduff and her son are slaughtered on the orders of Macbeth. Macbeth is showing his further descent into the darkness of his inner character as he strives to ensure his retention of the thrown. Macduff is away from lady Macduff on a quest to try to gather support for the overthrow of Macbeth.

Lady Macduff is angry because her husband left for England and because he did not tell her. She and her son are alone, and she is frightened. A messenger arrives to report that she and her son are in danger. Murderers arrive, and both Lady Macduff and her son are killed at Macbeth's order.

To put it bluntly, all are brutally killed by Macbeth's henchmen.  You see, once Macbeth obtains the crown, he becomes more and more obsessed with holding on to power.  As a result, he vows to eliminate anyone who he feels is a threat to him.  With that in mind, he knows that Macduff has gone into hiding, and Macbeth is afraid Macduff in plotting against him.  In an effort to draw Macduff back to Scotland (he is in England trying to convince Duncan's son Malcolm to join him in a rebellion against Macbeth -- he really is plotting against him), Macbeth dispatches assassins to kill Macduff's family.

The killing of Macduff's family is one of the more bloody acts in the play.  Lady Macduff begins the scene complaining that her husband has gone away and left her all alone -- she even suggests she might leave him and find another man to replace him.  Her son says he sees through her complaints and points out she still really does love Macduff.  After this exchange, Maceth's men swoop down upon them and kill them both -- first Lady Macduff and then her son (after she told him to run away).

So, by this point in the play, all of Macduff's family is dead -- murder by his kinsman turned enemy Macbeth.

In Act IV, Scene II of Macbeth, Lady Macduff is informed by Ross that Macfuff has gone to England. She says that to flee in this way makes him look like a coward and a traitor. "When our actions do not, / Our fears do make us traitors" (IV.ii.3-4). Ross tells her that Macduff knows what he is doing and he tells her that they are all under suspicion, "But cruel are the times when we are traitors / And do not know ourselves;" (IV.ii.18-19). 

Lady Macduff then tells her son that his father is dead. He disagrees. She tells him he is a traitor, explaining what a traitor is. Lady Macduff insists that traitors should be hanged. Even if Macduff is still alive, she believes he will be killed. She has lost complete faith in him. The dialogue between Lady Macduff and her son can seem distraught but also cold. Her son notes to her:

"If he were dead you'd weep for him. If you would not, it were a good sign that I should quickly have a new father." (IV.ii.61-63)

A messenger warns Lady Macduff to flee but she has done nothing wrong so she sees no point in fleeing. However, she doesn't heed Ross' advice that she too is under suspicion. The murderers arrive and kill them both. The son denies that his father is a traitor as he is killed. 

Lady Macduff feels abandoned. She is too stubborn or honorable to accept that she should also leave. Her son doesn't believe his father is dead and he even questions her loyalty, saying if she is not weeping, it is a good sign that he will have a new father. Therefore, the son remains loyal to his father to the end. 

In the overall plot of the play, this is an example of how Macbeth is able to cast suspicion from himself onto others. Here, by driving Macduff out of Scotland, he makes him (Macduff) look suspicious. Lady Macduff thinks that even if Macduff is not a traitor, his actions make him a coward. By refusing to leave, Lady Macduff shows strength of conviction but she also shows a certain stubbornness. 

Macbeth has at least four motives for ordering the murders of Lady Macduff and all of Macduff's children. Only one child, a boy, is actually shown, but subsequent dialogue makes it clear that there were many others. When Macduff is informed of the atrocity, he says:

All my pretty ones?
Did you say "all"? O hell-kite. All?
What, all my pretty chickens and their dam
At one fell swoop?   (4.3)

Macbeth is angry at Macduff for deliberately and flagrantly refraining from attending his coronation and thus ;publicly refusing to acknowledge him as king of Scotland. Then Macbeth is further incensed by Macduff's failure to attend the coronation banquet when all the other nobles are in attendance. Then in Act 4, Scene 1 Macbeth is informed that Macduff has fled to England, and he decides:

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.

He does this both for revenge and to make Macduff serve as an example of what will happen to the families, titles, and properties of any other thanes who might contemplate following Macduff's example. Macbeth's fourth and last motive for attempting to wipe out Macduff's family and his very existence and memory as a Scottish nobleman is that he was recently warned by the witches' First Apparition:

Macbeth! Macbeth! Macbeth! Beware Macduff!
Beware the Thane of Fife! Dismiss me. Enough. (4.1)