The poem's many metaphors (and similes) make its story more dramatic, and they also reflect the true nature (in the speaker's mind, at least) of the poem's characters.
In the second stanza, for instance, the speaker uses metaphors to portray his friend's beloved as malicious and lecherous. First, he says, "she crossed his path with her hunting-noose / And over him drew her net." This metaphor treats the woman like a hunter, and the speaker's friend as her prey. The speaker doesn't think they just met and were attracted to one another, but that this woman set out to ensnare him.
The speaker builds on this metaphor in the next stanza, saying that he doesn't want to see his friend added "To her nine-and-ninety other spoils." "Spoils" refers to ill-gotten gains—things stolen, plundered, etc. Now, the woman becomes like a pirate or mercenary, the speaker's friend no more than a trophy to add to her shelf.
In the speaker's mind, this woman is deceiving his friend, making him believe she's more interested in him than she really is. In lines 15-16, he explains that
An eagle's the game her pride prefers,
Though she snaps at a wren instead!
This metaphor again characterizes the woman as a hunter. The speaker thinks she'd rather chase after something as clever and powerful as herself—something as strong and fierce as an eagle—rather than scoop up an innocent little "wren." The speaker goes on to explain that he's "The eagle" because he's famous, while his friend "with his maiden face" (or youthful, beautiful features) is the wren.
After the speaker seduces his friend's mistress, his friend "eyes [the speaker] as the basilisk." A basilisk is a fearsome mythological creature whose gaze turns anyone who locks eyes with it to stone. A basilisk is also typically represented as a snake-like reptile, which hammers home the speaker's duplicity (snakes are common symbols of treachery and sneakiness): he has betrayed his friend. In doing so, he has metaphorically transformed his friend's "day to night." and "Eclips[ed] his sun's disk." That is, he's cast a dark, gloomy shadow over his friend's world, robbing him of the warmth and happiness represented by sunshine. The metaphor further subtly suggests that the speaker did this by making himself seem like the more attractive mate, dimming his friend's glow.
In lines 33-36, the speaker uses a simile that then expands into an extended metaphor:
[...] she lies in my hand as tame
As a pear late basking over a wall;
Just a touch to try and off it came;
'Tis mine,—can I let it fall?
The woman is like a happy pear soaking up the sun, ripe and dangling "over a wall." In short, she's tempting the speaker, who easily plucks her away from his friend. The ease with which this figurative pear falls off the branch ("Just a touch to try and off it came) implies that the woman was all too willing to go with the speaker—or, maybe, any man who came by.
The speaker is trying to make himself look better by insisting that the woman didn't need much convincing. He claims the pear is now his, suggesting that the woman has fallen in love with him. If he turns her away, it's like letting the pear "fall" to the ground, wasted.
The speaker builds on this metaphor in the following stanza, turning it into an extended metaphor:
With no mind to eat it, that's the worst!
Were it thrown in the road, would the case assist?
The speaker presents himself as a martyr here: he insists that he doesn't like pears (meaning, in terms of the metaphor, he has no interest in this woman), yet now she's in his "hand." He has no desire to stay with her, but he knows that if he doesn't—if he were to, say, toss her "in the road" like a piece of old fruit—it would only make the "case" against him seem even worse.
He then insults the woman further still, adding that this metaphorical pear was "quenching a dozen blue-flies' thirst / When [he] gave its stalk a twist." This suggests the woman was involved with lots of other men before he interfered with things. The image of him "twist[ing]" the "stalk" evokes his cruelty meddling in things that are of no concern to him.
There is one more metaphor in lines 47-48, as the speaker implores the reader to "think of [his] friend" before judging him too harshly, arguing that his friend "played" with "burning coals" his mistook for "bits of stone." That is, he's again arguing that his friend had no idea what he was getting into and that he was only looking out for him so he wouldn't get hurt.