In Macbeth, what happens to Fleance when Banquo is killed?
The stage directions in Act III, Scene III, arguably the climactic scene of the play, provide the answer to this question. After Banquo dies, "Fleance escapes." This is catastrophic for Macbeth, and marks the true beginning of his unraveling. The Second Murderer rightly notes that they "have lost / Best half of our affair" in failing to kill Fleance; they have been sent by Macbeth to kill Banquo and Fleance both, but, as the Third Murderer says, "the son has fled."
When the First Murderer tells Macbeth that Fleance has escaped, Macbeth says, "Then comes my fit again." He knows that having Banquo dead but Fleance, his issue, still alive does not really benefit him. He has orchestrated the murder of his friend and yet is still bound by "saucy doubts and fears." Macbeth knows that having Fleance killed was, indeed, more than "half our affair," because what the witches told Macbeth in their prophecy was not that Banquo would become king, but rather that Banquo's children would. Having Fleance still alive, then, means that he represents a continuing threat to Macbeth. It is this anxiety which drives Macbeth to seek the witches again to ask for clarification and further word on what will happen and what he should do.
To answer this question, take a look at Act III, Scene III. In this scene, Macbeth's henchmen lie in wait for Banquo and his son, Fleance. In the darkness, the henchmen attack Banquo, and during this scuffle, Banquo calls out to his son to run away:
Fly, good Fleance, fly, fly, fly!
Fleance listens to his father and escapes the scene.
Shakespeare does not tell us where Fleance goes. What's more important is that he has escaped, and his escape causes considerable anxiety for Macbeth. Remember that back in Act I, the witches told Macbeth that Banquo's sons would be kings. Fleance's escape, therefore, troubles Macbeth because he is now wondering if this prophecy will come true. Will Fleance be the cause of his undoing? Will Fleance become the new King of Scotland? This is a scenario which Macbeth is desperate to avoid.
According to the murderer who reports to Macbeth in Act III, Fleance has escaped being killed. In the earlier scene that dramatized the assassination of Banquo, one murderer says, "There's but one down; the son is fled." A second murderer replies, "We have lost best half of our affair." Macbeth is not pleased with this news. It was his intention to kill Banquo's son so that none of Banquo's heirs could ever rule Scotland.