The poem begins with an invocation. This is a brief call on the "muses"—ancient Greek goddesses—to grant the poet inspiration and creativity. Many, many famous poets began their work with invocations: John Milton, for example, asked the "heavenly muse" to help him compose Paradise Lost.
This invocation would thus feel pretty conventional—were it not for the fact that it's being made here, not by a standard-issue 18th-century white male poet, but by a young Black woman. Snatched from her family and enslaved when she was only a child, Phillis Wheatley was most likely the first Black woman to publish poetry in print anywhere in the world.
The allusion to the muses, then, signals that the speaker (generally taken as Wheatley herself) knows her poetry. It's clear that she is well-educated and has a passion for poetry: she describes herself as having an "intrinsic ardor" for the form, a God-given longing to write.
In other words, she's standing up for herself with this invocation, implying that she's just as inspired, and just as worthy to call on the muses, as any white male poet.
Alliteration between "prompts," "promise," and "pen" links the speaker's urge to write with the assistance of external forces and the act of writing itself. This connection subtly makes the point that the assistance of the muses need not solely belong to educated white men: this speaker, too, is "prompt[ed]" by divine inspiration. This has implications later on in the poem when the speaker compares her background to those of the students attending Harvard University (to whom this poem is addressed).