Why does Macbeth see Banquo as a threat in Act 3, Scene 1?
Since Macbeth's encounter with the witches, he cannot help but dwell on their prophecy. The witches prophesied that Banquo's descendants would take the throne one day, which means that Macbeth's position as the king of Scotland could be imperiled by Banquo and his child, Fleance:
Upon my head they placed a fruitless crown,
And put a barren sceptre in my gripe,
Thence to be wrench'd with an unlineal hand,
No son of mine succeeding. If 't be so,
For Banquo's issue have I filed my mind;
For them the gracious Duncan have I murder'd...
The witches' prophecy drives Macbeth to resort to some of the most brutal and desperate measures -- he decides to have Banquo and his child murdered. This will not only enable him to be at peace, but it will also ensure that his quest to become the king of Scotland was not futile. He certainly does not want to fathom that killing Duncan and sacrificing his inner peace could amount to nothing one day. He wants to ensure that he would be untouchable as the ruler of Scotland and that no one could jeopardize his privileged position.
The witches predicted that Banquo's descendants would become kings; therefore, Macbeth fears that these descendants may take his place. He says in Act 3, scene 1, lines 70-71: "For Banquo's issue [descendants] have I filed [defiled] my mind;/For them the gracious Duncan have I murdered...." If Banquo lives (or his son Fleance, for that matter), then Macbeth thinks he has killed the king only so that Banquo's children/grandchildren could perhaps wear the crown. Macbeth has no children, no one he can name as heir to the throne. Only Banquo now stands in the way of his happiness and security, Macbeth believes. To eliminate this threat, he thinks he must kill Banquo---and Fleance.