This famous quote opens Act III, and it is important to realise how vital this quote is. Banquo has silently witnessed what has transpired in Act II and how Duncan has been assassinated and Macbeth has seized power. He alone was there with Macbeth when the witches predicted that Macbeth would become Thane of Cawdor, Glamis, and then King, and now we see that he suspects Macbeth of having gained these prophesied titled through foul means, as Banquo says tha the fears that Macbeth "play'dst most foully" for the titles he has won. This is a very important soliloquy in the play therefore, as in it Banquo voices the suspicions that he has of Macbeth and internally debates the truth of the prophecies of the "Weird Women." For Banquo, too, received prophecies, and surely if the prophecies that applied to Macbeth came true, then those relevant to Banquo will come true as well. However, as that involved one of Banquo's heirs becoming King, this is potentially a treacherous thought, so he is quick to be silent when he hears other characters enter.
This quote also shows why Macbeth fears Banquo and wants him dead. Banquo is intelligent, suspicious, and dangerous. He is sure that Macbeth murdered King Duncan, and Macbeth is well aware that Banquo is sure of his guilt. Macbeth, however, does not know what Banquo might do about it. Banquo has all the qualities Macbeth ascribes to him in one of his own soliloquies. Banquo is the one man whose being Macbeth fears. He is probably afraid that Banquo could organize a plot against him because the man possesses leadership qualities among other things. Macbeth is like Claudius. Macbeth is wondering what Banquo is thinking and planning, while Claudius is wondering the same things about Hamlet. Banquo is wise enough to hide his thoughts and feelings from everyone, including Macbeth. He is treating Macbeth with the utmost courtesy and respect. It is obvious to both men that Banquo would be closer to realizing the witches' prophecies with Macbeth out of the way. Banquo's offspring would be closer in line of succession to the throne.