Stanza 1 Summary

Get out the microscope, because we’re going through this poem line-by-line.

To sing of Wars, of Captains, and of Kings,
Of Cities founded, Common-wealths begun,

Woah, Bradstreet starts out on a big note here. It's like she's ringing a gong to announce her presence. Her speaker leaps right into the grandest subjects of history: war and politics. If you were just reading this first line, you might think she (and we're just assuming that the speaker is a she, since we have no evidence to the contrary) was about to write an epic poem, like The Iliad.
For a poem entitled "The Prologue," this seems like kind of an odd move. Shouldn't we be introduced gradually to the book? (Prologues come first in books, usually to set the scene by the way.) We wonder if Mrs. Bradstreet's going to be tricky with us all through this poem.
On a totally unrelated note, this line makes us think of one of our favorite goofy poems, Lewis Carroll's "The Walrus and the Carpenter": "'The time has come,' the Walrus said,/ 'To talk of many things:/ Of shoes--and ships--and sealing-wax-- /Of cabbages--and kings.'"
We're not sure Anne B. would have been a fan of the comparison, but we couldn't resist.

For my mean Pen are too superior things;

All of a sudden, she switches it up on us. Now she tells us this poem isn't really going to be about these big events and famous people at all. Apparently she feels like these grand historical themes are "too superior" for her "mean" (that means "low" or "unimportant") pen.
In this case, she doesn't mean that the actual pen she's using isn't any good. She's using it symbolically to represent her skill as a writer. That kind of symbolism, where you take a part of something to stand in for the whole, is called metonymy.

Or how they all, or each their dates have run,
Let Poets and Historians set these forth.

She'll leave the big work of describing the past and how it took place ("how […] their dates have run") to "Poets and Historians." Apparently she's going to let those guys talk about (or "set forth") those topics, while she deals with something less major. That's an… interesting way to put it.
It also seems pretty significant that she refers to other people as "Poets," which implies that she doesn't include herself in that category. In this opening stanza, it almost feels like she's trying to hide from who she is.
Now that you've seen a few lines, maybe you've picked up on the steady rhythm of this poem. It's written in pretty much perfect iambic pentameter. If that term's got you scratching your noggin (and, you know, you've washed your hair recently), then head on over to the "Form and Meter" section for a full breakdown.
While we're at it, let's check out the rhyme too. Each stanza in this poem has the same rhyme scheme. The first line rhymes with the third, the second line rhymes the fourth, and each stanza ends with a pair of rhyming lines (we call those last two a rhyming couplet). So if we wrote out the general pattern using letters, it would go ABABCC (where each letter represents the sound of that line's end rhyme).
Having fun, yet? Alrighty, then, let's move on…

My obscure lines shall not so dim their worth.

This speaker won't let her "obscure" lines mess with the most important material or history, because she thinks that if she does it badly it will make them seem like they are worth less than they are.
But wait a minute. Let's look at this line again. Check out the word "obscure." That can mean "not important," "humble," "not famous." But it can also mean "dark" or "hidden by darkness." So when she tells us that her obscure ("humble") lines might "dim" history, she's playing on the double meaning of "obscure."
Maybe that doesn't seem so amazing, but we think it's important, because it gives us a hint that Bradstreet is playing two games at once here. On the one hand, she's telling us that her poetry isn't good enough to let her play with the big boys, and on the other, she's showing off her poetic chops. Let's keep an eye on this lady, folks—she's got some tricks up her sleeve.