Concerning Shakespeare's Macbeth, specifically, when Macbeth talks about it being one thing to be king, and another all together to be safely king (the "safely thus" quote), he is revealing why he orders Banquo to be killed: Banquo knows about the witches' predictions.
Macbeth has to eliminate Banquo, because Banquo can finger him, so to speak, for the killing of Duncan. He suspects Banquo suspects him (and he's correct, of course), and wants to eliminate him before Banquo says anything. That's what will make Macbeth feel safe, he thinks.
When Macbeth talks about Banquo's character traits, he is demonstrating that Banquo is different from him. Banquo will not give into temptation as Macbeth has. Banquo is a character foil to Macbeth. He is everything Macbeth isn't. Since he will not act treacherously as Macbeth does, he is a threat to Macbeth.
Unfortunately, Banquo is also naive. He chooses not to say anything about the witches' predictions, is on the wrong end of dramatic irony when Macbeth questions him about his ride and whether or not Fleance will be riding with him, and, instead of riding away when he has the chance (and one can assume he could have escaped numerous other times), is returning to the castle for Macbeth's banquet when he and Fleance are attacked. He is naive and far too trusting of the ruthless Macbeth.
Fleance, of course, is the heir that must be destroyed, according to the witches' predictions. Killing him, without killing Banquo, wouldn't have been enough, of course, since Banquo could have had more children if he'd have lived. More importantly, notice that at the time Macbeth killed Duncan, being king was enough. At the time of the predictions, Macbeth gives no thought to the idea that he will not create a dynasty but only be king himself. Macbeth's ambition is such, however, that once he achieves the crown, the crown, for himself alone, is no longer enough. His ambition is so deep that once he achieves the crown, he wants more--he wants a dynasty.
In short, the most important reason Macbeth orders Banquo's death is that Banquo knows about the witches' predictions. Banquo can nail him for Duncan's murder. He orders Fleance's death because Fleance is Banquo's heir. Anything else is secondary.
To be thus is nothing,
48 But to be safely thus. Our fears in Banquo
49 Stick deep; and in his royalty of nature
50 Reigns that which would be fear'd. 'Tis much he dares;
51 And, to that dauntless temper of his mind,
52 He hath a wisdom that doth guide his valour
53 To act in safety. There is none but he
54 Whose being I do fear: and, under him,
55 My Genius is rebuked; as, it is said,
56 Mark Antony's was by Caesar.
I agree that Macbeth's soliloquy in Act III, scene i demonstrates Macbeth's fear of Banquo. We must look specifically at how Macbeth views Banquo -- he sees his "royalty of nature" -- do not forget that the Witches have prophesized that Banquo will not be a king, but instead beget kings; Macbeth wants to kill Banquo for the "royalty of nature" - the literal lineage that will come about if Banquo lives, and for Banquo's goodness. Banquo is Macbeth's character foil; he demonstrates all the qualities that Macbeth doesn't -- goodness, kindness, honesty, and checked ambition.
Macbeth also fears Banquo's intelligence. It is not by chance that Shakespeare sets up an insight into Banquo's thoughts before Macbeth demonstates his suspicions:
1 Thou hast it now: King, Cawdor, Glamis, all,
2 As the weird women promised, and I fear
3 Thou play'dst most foully for't; yet it was said
4 It should not stand in thy posterity,
5 But that myself should be the root and father
6 Of many kings. If there come truth from them—
7 As upon thee, Macbeth, their speeches shine—
8 Why, by the verities on thee made good,
9 May they not be my oracles as well,
10 And set me up in hope? But hush, no more.
Banquo is astutely aware, a "wisdom that guide[s] his valor," as can be compared to Marc Antony and Caesar; Banquo knows that Macbeth "play'dst most foully for't" and knows that Macbeth will listen to the remaining prophecy -- this is why he tries to ride away with Fleance. Macbeth is aware of how Banquo would think, since they both bore witness to the prophecy. It is, essentially, psychological warfare; who can beat whom to the punch?
Once Macbeth becomes king he is concerned for his legacy. If he believes the witches' predictions (which he clearly does), he understands Banquo will be the father of a king. That makes Fleance a definite threat to Macbeth's position as current king. But it's Banquo he wants dead more than the potential heir.
The reason Macbeth feels he must kill his former friend is alluded to in the explanation for your question. I'd suggest that Macbeth's paranoia causes him to fear Banquo himself. Banquo is the only person other than Lady Macbeth who could know with any certainty that Duncan was not murdered by his sons. Your teacher says it a different way--that Banquo is a man of good character and may soon reaveal Macbeth's murderous act in order to spare the country and punnish the king-killer.
In the end, it's a combination of things which cause Macbeth to murder Banquo. Specifically, if all he were concerned about was his legacy, he'd only have killed Fleance. Instead, he kills both father and son. In fact, as he's contracting with the murderers, he has to "sell" them on killing the well-known Banquo--quite possibly because he was a man of good reputation and morality. He describes them to the murderers as "this good man and his issue" (Fleance). It's not until the end of the planning when Macbeth asks them to kill them both, not just Banquo. Banquo was the first and primary target.
The most significant evidence of your teacher's position is what Macbeth says to himself (IIIi), with no one else to hear, after the arrangements have been made:
No mention of Fleance in any form or fashion, but an inference that Banquo would be heading to heaven rather than hell. It's clear he wanted Banquo killed; Fleance is almost an afterthought.
You best bet would be to look more closely at Macbeth's soliloquy in Act 3, scene 1:
To be thus is nothing, but to be safely thus--
In this soliloquy Macbeth lists all the reasons he thinks Banquo should die. He is afraid of Banquo because of his "royalty of nature." He has a "dauntless temper" [courage]; he has "a wisdom that doth guide his valor." It is interesting that Macbeth lists these noble characteristics--courage and wisdom-- first before he comes to the point about Banquo's sons being kings. Perhaps he considers these more important. Macbeth fears Banquo; he fears Banquo might suspect him of murder and will not be afraid to act on his suspicions. He fears Banquo's goodness now that he himself has lost his "eternal jewel" [soul].
macbeth kills banquo towards his actions on the phrophecies
he belives that what the witches said was the right thing
so he belives that he has to kill him in order to become king or else its fleance. banquos sons who would become king
this is very brief i hope it is enough