Macbeth hires three murderers to kill Banquo and his son, Fleance, because the Weird Sisters told Banquo that he would "get kings, though [he will] be none"; in other words, he would father kings, though he would never be a king himself (1.3.68). Macbeth is childless, and, after he murders Duncan, it occurs to him that he has saddled his conscience and defiled his soul all so that Banquo's descendants might reign in Scotland. This makes him angry and bitter, and so he determines to kill Banquo, so that he can father no more children, and Fleance, Banquo's only child, so that he can never take the throne or have children of his own who might do so.
Fleance escapes a particularly bloody and sad scene, his father yelling at him to "Fly, good Fleance, fly, fly, fly! / Thou may ’st revenge--" (3.3.19-20). The danger Fleance's continued life poses to Macbeth is the result of the Weird Sisters' prophecy to his father. Macbeth loathes the idea that he has done everything to gain the crown just to lose it to someone else's sons, and this is the potential conflict for him. In actuality, he will die before any of Banquo's issue can take the throne. However, as the other commenter mentioned, the king on the throne when Macbeth was first performed was James the I of England (James the VI of Scotland), and he could trace his lineage back to Fleance and Banquo. Thus, the Weird Sisters were right!
Fleance escapes and the conflict this could cause in the future is that Fleance knows that Macbeth killed his father and he could possibly tell someone and have Macbeth excecuted.