The poem begins with the speaker addressing someone "dear[]" to her. (Note that the speaker doesn't need to be interpreted as a woman, but for clarity's sake, we will be referred to as "she/her" throughout the guide).
This loved one, the speaker says, should "Sing no sad songs" in the event of the speaker's death. It isn't clear why the speaker is thinking about death, but what is immediately clear is her lack of sentimentality on the subject. She doesn't want her loved one to feel obligated to mourn her.
Ironically enough, the speaker uses a song to implore her "dearest" not to sing for her once she's died, almost as if she is mourning her own death in advance. Perhaps, then, she is a little sentimental after all!
Given that this poem is a "song," it makes sense that its language is rhythmic and musical. These lines are written in ballad meter, a bouncy meter made up of alternating lines of iambic tetrameter (four iambs, feet with a da-DUM rhythm, per line) and trimeter (three iambs per line):
When I | am dead, | my dearest,
Sing no | sad songs | for me;
The meter isn't totally regular, however, as readers can see right away here. For one thing, the speaker leaves off the final stressed beat in line 1, a variation she will return to throughout the poem. Still, there's a clear, steady rhythm to these lines.
The consonance and more specific sibilance of these lines add yet more music still. Note all the repeated /s/, /d/, /m/, and /ng/ sounds:
When I am dead, my dearest,
Sing no sad songs for me;