How does Shakespeare make Macbeth a character with whom the audience can sympathize?
Shakespeare is notoriously ambiguous in his tragedies, which contributes to their popularity among teachers and scholars, since they can be debated relentlessly. The last I heard, for instance, more commentary had been written about Hamlet than any other book in existence, except the Bible.
Macbeth is ambiguous, as well. For instance, many productions of the play show Banquo's ghost in Act 4.3 at the feast. The ghost's appearance seems likely to be real, or actual, in a play filled with the supernatural. Shakespeare uses a ghost in Hamlet, why not in Macbeth? Also, Gertrude in Hamlet doesn't see the Ghost of King Hamlet during the bedroom scene when the Ghost is seen by Hamlet, so there is nothing unusual about no one else seeing Banquo's ghost except Macbeth.
And if the ghost is real, then Macbeth is not insane. I think we misinterpret when we place too much emphasis on Macbeth's so-called insanity, and relieve him of responsibility by doing so. If Macbeth is not responsible, it is because of an overall design by fate or predestination, an issue important in Elizabethan England due to the Protestant Reformation, which brought the issue to the attention of Europeans.
That said, any sympathy for the character of Macbeth the audience feels must come from some other source than his insanity. And it must not come from feeling that he isn't responsible--he is. The witches tell him only that he will be king: he turns that into the thought that he needs to be king now! He turns the prediction into the need to kill Duncan. Also, he seeks the witches out in Act 4.1, not the other way around. He also makes the mistake of cutting his wife out of the decision-making process once Duncan is dead: he kills the grooms, orders the murders of Banquo and Fleance and Macduff's family, all on his own. These are all mistakes he makes, without anyone else's help.
So where does sympathy come from? His nobility in defending Duncan at the beginning of the play; his nobility in death at the close of the play; his hopeless situation once he is surrounded--he is like a bear chained to a tree, attacked by a pack of dogs for the entertainment of an audience (Act 5.7); from the intellectual side of his personality that recognizes everything he's done is meaningless (the "Tomorrow" speech, Act 5.5).
Sympathy for Macbeth is not in any way central to the play. His ambition and fall from grace are central. But if one does feel sympathy for him, it comes from the above. He allows himself be deceived by the witches and talked into Duncan's killing by his wife (and these are conscious decisions). And he takes over from there and brings about his own downfall. That cannot be dismissed by any insanity. He is guilty. But sympathy can exist for him, anyway. Not because we take the guilt away, but because we see other characteristics than just evil.
In Shakespeare, in sophisticated literature, characters are mixtures of positive and negative character traits, as are actual human beings. We don't need to make excuses for Macbeth in order to feel sympathy for him.
A reader's sympathy for Macbeth is mainly derived from Macbeth's mental instability in the play. His hallucinations and apparitions reflect his mental collapse. The first important evidence of this is his soliloquy in Act II, Scene 1, when he visualizes a dagger before he murders Duncan.
Is this a dagger which I see before me,The handle toward my hand? Come, let me clutch thee. I have thee not, and yet I see thee still.
In addition, later in the play, he sees Banquo's ghost sitting in his chair at a feast he is hosting when he is King. This occurs in Act III, Scene 4. More importantly, Lady Macbeth does not see Banquo's ghost, and Macbeth is surprised that she is not afraid.
You make me strange
Even to the disposition that I owe,
When now I think you can behold such sights,
And keep the natural ruby of your cheeks,
When mine is blanch'd with fear.
The reader can also see Macbeth's ill mental health in his obsessive thoughts of the witches' apparitions. He insists on knowing what will happen in the future, and visits them to demand a response. The witches, lead by Hecate, set out to confuse Macbeth in their prophecies to him in Act V, Scene 1. Due to Macbeth's instability, he can not truly be responsible for his irrational behaviour.
If there is any hope of identification with Macbeth, it probably lies in the beginning or the outset of the play. On one hand, he is presented as a valiant and brave leader on the battlefield. The notion of ambition seems to be quite vague for him, until his wife begins to tap into this sensibility that might have been more dormant than anything else. In no way does this absolve him of all that he does, but in terms of audience sympathy, there is a level of empathy because the notion of being "geared" or "manipulated" into a position to do what one knows should not be done is an element to which the audience can feel some level of sympathy and understanding.