How did Macbeth win the respect of King Duncan?
In short, Macbeth in Shakespeare's play of the same name, wins Duncan's respect and is rewarded by Duncan when, as the "bloody man" or Captain says:
...he [Macbeth] unseamed him [the traitor Macdonwald] from the nave to th'chops,
And fixed his head upon our battlements. (Act 1.2.22,23)
In other words, Macbeth disembowels the treacherous Macdonwald, beheads him, and raises his head on a pole. Macbeth also repels the Norwegian force that is allied with the Thane of Cawdor, matching Cawdor blow for blow until, finally, Duncan's forces win.
Thus, Duncan assumes Macbeth to be his loyal Thane, and, along with his respect, gives him the Thane of Cawdor's castle and lands.
Ironically, at the close of the play, Macbeth will be the one beheaded for his treachery.
I think that the answer you are looking for is that Macbeth wins Duncan's respect through his bravery in battle.
I assume you are talking about what is said in Act I, Scene 2. In that scene, we see Duncan talking about Macbeth. He calls Macbeth "valiant" and "worthy."
The reason that Duncan says these things is because of Macbeth's performance in battle. Duncan has just been told that Macbeth has performed very bravely in a battle against people who would have overthrown Duncan. Specifically, Duncan has just heard about how Macbeth killed Macdonwald.
It is in Act I, Scene II of Macbeth that we learn the reason for King Duncan's respect. Macbeth is a brave warrior whose fighting prowess had led to a great victory on the battlefield. Shakespeare emphasizes this point through his use of language. He likens Macbeth and Banquo to an "eagle" and a "lion," for instance, which suggests their physical strength and prowess. Moreover, it is reported that Macbeth "doubly redoubled strokes" upon his enemy. This scene is likened to "Golgotha," the name given to the place of Jesus' crucifixion. By characterizing Macbeth in this way, Shakespeare highlights the respect and admiration that other men now have for him.
In response, King Duncan rewards the "noble" Macbeth by making him the Thane of Cawdor. This not only fulfills the witches' prophecy, it also encourages Macbeth's ambition which, rather ironically, leads to King Duncan's own demise.