How do supernatural occurrences play an important role in Macbeth?

The importance of the supernatural in Shakespeare's Macbeth is best viewed in terms of Shakespeare's intended audience. In Elizabethan and Jacobean England, there was widespread belief in black magic and the supernatural. Many people in Shakespeare's audience believed that supernatural beings like the Witches actually existed, and that they interacted with humans, caused disease, and spread chaos throughout the country.
Perhaps the most important person in Shakespeare's audience was James I, King of England, for whom scholars believe Shakespeare wrote Macbeth. James I considered himself an expert on witchcraft and wrote a book, Daemonologie, published in 1597, which explored various aspects of divination, witchcraft, and black magic.
Shakespeare's audience believed that the supernatural events depicted in Macbeth could happen. Some people in our modern world believe in the supernatural, and even those who don't would willingly suspend their disbelief to enter the world of the play and imagine what it would be like to be one of the characters in Macbeth or a member of Shakespeare's audience watching the play who did believe in the supernatural.
The supernatural in Macbeth represents the unknown and, more significantly, the fear of the unknown, which Shakespeare used to engage the audience in the play, build anticipation, frighten them, and bring the events of the play into sharp focus in their minds.
The first scene in the play is devoted exclusively to the Three Witches. Thunder and lightning precedes their entrance, and even though the Witches don't actually do anything supernatural or even act much like witches, Shakespeare sets the tone and the supernatural environment for the play. Through the Witches, Shakespeare invites the audience into the world of the play and establishes a sense of foreboding and anticipation for what's to come.
Two scenes later, in act 1, scene 3, the Witches are back again, and at first they're by themselves. This time, they're acting more like witches, and they're talking about witchy kinds of things.
Soon Macbeth and Banquo enter, the Witches appear to them, and the Witches make specific and vague prophecies to them—specific to Macbeth, and vague to Banquo.
To Macbeth they say,

SECOND WITCH: All hail, Macbeth! hail to thee, Thane of Cawdor!
THIRD WITCH: All hail, Macbeth, that shalt be King hereafter! (1.3.51–53)

To Banquo they say,

FIRST WITCH: Lesser than Macbeth, and greater.
SECOND WITCH:Not so happy, yet much happier.
THIRD WITCH: Thou shalt get kings, though thou be none. (1.3.68–70)

These few, simple prophecies set into motion all of the events that follow in the play.
The Witches don't return until act 3, scene 5, when the head witch, Hecate, chides them for what they've been doing without her:

FIRST WITCH: Why, how now, Hecate? You look angerly.
HECATE: Have I not reason, beldams as you are,Saucy and overbold? How did you dareTo trade and traffic with MacbethIn riddles and affairs of death;And I, the mistress of your charms,The close contriver of all harms,Was never call'd to bear my part,Or show the glory of our art? (3.5.1–9)

The Witches still haven't done anything particularly supernatural except make somewhat vague prophecies to Macbeth and Banquo, but Hecate tells the Witches what's going to happen now that she's back in charge.
First, she tells the Witches to get their witch things organized to meet Macbeth:

HECATE: Get you gone,And at the pit of AcheronMeet me i’ the morning. Thither heWill come to know his destiny.Your vessels and your spells provide,Your charms and every thing beside. (3.5.14–19)

Then Hecate tells them what she's going to do, and what's going to happen when they meet Macbeth:

HECATE: I am for the air; this night I'll spendUnto a dismal and a fatal end.Great business must be wrought ere noon:Upon the corner of the moonThere hangs a vaporous drop profound;I'll catch it ere it come to ground.And that distill'd by magic sleightsShall raise such artificial spritesAs by the strength of their illusionShall draw him on to his confusion.He shall spurn fate, scorn death, and bearHis hopes ’bove wisdom, grace, and fear. (3.5.20–31)

The next time we see the Witches is in act 4, scene 1, the Apparition scene. The audience has been watching the events foretold in the prophecies unfold just like the Witches said they would, and this is where Shakespeare gives the audience the supernatural payoff scene that they've been anticipating.
The Witches and Hecate are getting ready to meet Macbeth, and they're involved in some serious witch business with a bubbling cauldron, snakes, newts, frogs, bats, dogs, lizards, dragons and all kinds of other spooky, supernatural things.

WITCHES: Double, double, toil and trouble;Fire burn and cauldron bubble. (4.1.10–11)

Then the Witches don't simply conjure up ghosts, like those that appear to Hamlet or Richard III, which could be in their own minds. The Witches conjure up apparitions, completely new and different supernatural beings who convey new information to Macbeth.

SECOND APPARITION: Be bloody, bold, and resolute; laugh toscornThe power of man, for none of woman bornShall harm Macbeth. [4.1.86–90]
THIRD APPARITION: . . . Macbeth shall never vanquish’d be untilGreat Birnam Wood to high Dunsinane HillShall come against him. (4.1.103–105)

Then, to top it off, the Witches bring in the ghost of Banquo to make it seem more immediate and real to the audience and to frighten Macbeth and the audience.

MACBETH: Thou are too like the spirit of Banquo. Down! (4.1.125)

Then, after the Apparition scene,

[The witches dance and then vanish, with Hecate]

The Witches and Hecate are never seen again in the play.
What remains is for the audience to watch how the supernatural apparitions' information to Macbeth plays out, to Macbeth's ultimate, tragic demise.

The Three Witches, also referred to as the Weird Sisters, presumably have supernatural powers, allowing them to generate prophecies. The play opens with the witches, who tell Macbeth that he will be king. This highlights his already established ambitions, encouraging him to kill Duncan and foreshadowing future events in the play. When Macbeth seeks the witches out in act 4, the Weird Sisters have been ordered by Hecate, the leader of the witches, to deceive him into thinking he is untouchable as king. This spells out his doomed fight with Macduff in the final act.
In act 2, Macbeth sees a floating dagger, the same weapon used to kill Duncan. This apparition represents Macbeth's murder, which weighs on his conscience. In act 3, the ghost of Banquo appears to Macbeth and cannot be seen by Lady Macbeth, who is also in the room. This raises the question of whether the ghost is real, presenting itself only to Macbeth, or Macbeth's psyche manifesting his evil come to haunt him. Similarly, in act 5, Lady Macbeth sees blood on her hands, representing the role she played in Macbeth's murderous acts and their stain upon her conscience. The supernatural elements in Macbeth contribute to the play's sinister and deathly atmosphere and reinforce the theme of fate.