Poetry, like any art form, can change the way people think and feel. It does this not by presenting an argument but by creating a strong emotion within the reader. In the terminology of classical rhetoric, this is known as a pathos appeal, an appeal to the reader or listener's emotions. When used unskillfully, a pathos appeal can sometimes backfire, coming across as cloying and manipulative rather than persuasive. When used by a great poet, though, a pathos appeal in a poem can profoundly impact a reader.
A poem that comes to mind for me here is Mark Doty's "The Embrace," which I have linked to below. The poem is an apostrophe, delivered by the speaker to his partner, who has died of AIDS. The speaker has had a dream of his partner, and he recounts details with tender attentiveness: "your face, the physical fact of your face: / inches from mine, smooth-shaven, loving, alert." The poem ends with this heartbreaking stanza:
Bless you. You came back, so I could see you
once more, plainly, so I could rest against you
without thinking this happiness lessened anything,
without thinking you were alive again.
The poem is very poised, never too forthright about the speaker's emotions but rather presenting details and letting the reader feel those emotions instead. This last stanza, though, is the most openly emotional the piece gets. Doty's earned that emotion, and it hits like a gut punch.
To return to the original question, how might this poem represent a moral and political force? Personally, I don't know how anyone could come away from this poem unmoved. No matter your sexual orientation, you can feel the speaker's joy in seeing his partner again, and you can feel the bittersweet realization that this reunion didn't actually happen. Someone with negative feelings toward LGBTQ people may—it's not a guarantee—see something humanizing and relatable about this speaker, something universal in his experience. And if this reader's perception changes, they may be inclined to, for instance, vote for someone who supports the LGBTQ community. Again, this emotional response may not always, or even often, occur. But if the reader is receptive, a piece as powerful as Doty's may leave a powerful impression.