Macbeth basically convinces the murderers to kill Banquo and Fleance by making it seem like Banquo is their common enemy. For one thing, the murderers have to obey Macbeth as their king. However, Macbeth goes further by trying to make it seem like the murderers have some motivation of their own to want Banquo dead.
Macbeth questions whether the murderers would object to killing a man and his son if the man has mistreated them (the murderers). He calls Banquo one
Whose heavy hand hath bowed you to the grave
And beggared yours forever
Macbeth implies that Banquo is to blame for the murderers's poverty and virtual slavery to work. He tries to win them over by saying that Banquo doesn't care about them, so why should they care about him? Banquo's rule over these men means that they and their sons will never prosper.
Macbeth then tries to appeal to the murderers's reply that they are "men" by saying that yes, they are men, so they are not dogs. They should act like men and not allow themselves to be treated as lesser. Macbeth also refers to Banquo as "your enemy." The men reflect on the sorry state of their lives and agree that Banquo is their enemy, to which Macbeth replies that he is also the enemy of the king. He finally tells the murderer that he (Macbeth) cannot commit the murder himself, as he must be at the feast with mutual friends and cannot have suspicion cast upon him.
The murderers do agree to complete this mission, though Banquo's son Fleance escapes. The survival of Banquo's son continues to trouble Macbeth as he descends into madness.