Macbeth has only recently become King of Scotland after murdering King Duncan, but he decides that in order to keep the throne, and his life, he must murder Banquo and his son, Fleance, to avoid the prophecy that the witches made to Banquo in act 1, scene 3, "Thou shalt get [beget] kings, though thou be none" (1.3.70).
To this end, Macbeth enlists the services of two murderers. Macbeth tells the murderers that, as king, he could simply order Banquo executed, but there are "certain friends that are both his and mine" (3.1.132) that he dare not offend so early in his reign by doing so. Instead, Macbeth hopes to keep the business of killing Banquo out of "the common eye" (3.1.136) by employing the two murderers.
It's unclear why Macbeth chooses these two particular murderers, with whom he apparently has some unfortunate history to overcome, rather than any other two murderers in the entirety of Scotland who don't have to be convinced to murder Banquo and Fleance and who wouldn't hesitate to do the deed simply at Macbeth's order or request.
Macbeth had a meeting with these two murderers just the day before in which he set the groundwork for asking them to kill Banquo and Fleance. In this second meeting, Macbeth hopes to raise their anger and resentment toward Banquo to such a level that they won't hesitate to do what Macbeth requests of them.
Macbeth blames the murderers' lack of promotion and advancement on Banquo—"it was he...which held you so under fortune" (3.1.82–83)—although it's likely that Macbeth, not Banquo, interfered with their promotions. Macbeth tries to incite the murderers to consider Banquo their utmost enemy and to take revenge against him for what they believe he's done to them. However, at the same time that Macbeth is trying to solicit their services, he seems to insult them by comparing them to dogs and by casting aspersions on their manhood—in much the same way that Lady Macbeth did to Macbeth when he originally decided not to murder Duncan in act 1, scene 7. It worked for Lady Macbeth then, and it works for Macbeth now.
It appears, however, that the two murderers don't really need much convincing to murder Banquo. Both of them tell Macbeth that they're so down on their luck that they would do anything that Macbeth asks them to do, even at the risk of their own lives.
Macbeth gives them the job of killing Banquo and Fleance and sends them away, telling them that he'll come to them within the hour with specific information about Banquo's location so that they can commit the murders that very evening, "for ’t must be done tonight" (3.1.145).
When the time comes for the murders of Banquo and Fleance (in act 3, scene 3), a third murderer appears who is unknown to the other two murderers. The Second Murderer tells the First Murderer that they shouldn't mistrust the Third Murderer since he says he was sent by Macbeth, and he has the same orders from Macbeth that they do. They fail to realize, however, that it's Macbeth who doesn't trust them to carry out the murders, and he's sent a third murderer to ensure that the murders are committed "with no rubs or botches in the work" (3.1.148).
Nevertheless, even three murderers still manage to botch the work. They fulfill Macbeth's order to kill Banquo, but Fleance escapes.