From the outset, the witches' predictions have a marked effect on Macbeth, so much so that Banquo notices Macbeth's response during their encounter with them. Banquo remarks, in Act 1, scene 3:
Good sir, why do you start; and seem to fear
Things that do sound so fair?
It is clear at this point that Macbeth finds credence in the witches' predictions that he will become Thane of Cawdor and that he shall be king. Banquo's comment seems to indicate that Macbeth has a fright on hearing these words, probably because the witches have so accurately reflected his own 'deep and dark desires' - his ambition to be king. In the same scene Banquo also tells Angus and Ross:
Look, how our partner's rapt.
It is clear that Macbeth has been enraptured by what the witches have said. Banquo later also reflects on the effect that the predictions have had on Macbeth when he muses in Act 3, scene 1:
Thou hast it now: king, Cawdor, Glamis, all,
As the weird women promised, and, I fear,
Thou play'dst most foully for't:
It is clear that Banquo suspects Macbeth of murdering King Duncan. He suggests that Macbeth had used foul means to achieve what the witches had predicted.
Macbeth is clearly overwhelmed by what the witches predict. On being informed by Ross that he had been given the title thane of Cawdor, he responds as follows in an aside:
Glamis, and thane of Cawdor!
The greatest is behind.
He clearly believes that the greatest hurdle to his 'overriding ambition' has been overcome and that it would be easy to achieve his goal. This is further emphasized when, again in an aside, he comments:
Two truths are told,
As happy prologues to the swelling act
Of the imperial theme.
Macbeth is clearly ecstatic that he now holds two titles and he believes that they are precursors to his holding the highest title of all, that of king of Scotland.
So overjoyed is Macbeth by the predictions and the positive result, he believes stems from them - becoming thane of Cawdor, that he immediately forwards a missive to his wife informing her of the grand news. He asks her 'to lay it' to her heart. Macbeth is certain about achieving the crown.
When he arrives at his castle later, he and Lady Macbeth immediately start plotting Duncan's murder. Macbeth is plainly spurred on by the predictions and although he expresses some doubt about the plot later, he is easily persuaded by his wife to continue.
Macbeth has so much faith in the accuracy of the witches' predictions, that he, at the pinnacle of his tyranny, seeks them out to provide him with more assurances. The witches tell him that,
none of woman born shall harm Macbeth
Macbeth shall never vanquish'd be until
Great Birnam wood to high Dunsinane hill
Shall come against him.
Macbeth takes the witches at their word and believes that he is practically invincible since the two predictions state propositions which are naturally impossible. So confident is he that he states:
That will never be
Who can impress the forest, bid the tree
Unfix his earth-bound root? Sweet bodements! good!
Rebellion's head, rise never till the wood
Of Birnam rise, and our high-placed Macbeth
Shall live the lease of nature, pay his breath
To time and mortal custom.
Macbeth does not realise that he has been deceived by equivocation and double-speak and he relentlessly continues on his destructive path towards his own doom.
The witches' prophecies served to spur Macbeth's ambition, the driving force of the conflict in the play. When Macbeth initially meets the witches, they tell him he will be Thane of Cawdor and King. Macbeth dismisses this at first, but once he becomes Thane of Cawdor, he takes stake in what the witches say for the remainder of the play. Had the witches not propheseyed Macbeth being king, the thought would likely not have entered his mind. The witches also plant conflict between Macbeth and Banquo, driving Macbeth to have Banquo killed, thus increasing his guilt and mental instability. In the witches' final prophecies, they make Macbeth feel invincible, leading him to believe that he cannot be overthrown despite plots against him.
The "Macbeth Summary" portion of the link below touches briefly on this topic as well.
This question has already been answered numerous times on eNotes. Here is a comprehensive list to get you started: http://www.enotes.com/macbeth/q-and-a/tags/prophecies