There's a part in Romeo and Juliet where Juliet appears at her window while Romeo is below. When he sees her, he starts addressing her: "But soft, what light through yonder window breaks? It is the east and Juliet is the sun. Arise fair sun and kill the envious moon who is already sick and pale with grief." If you've ever heard that speech, or any speech where somebody is addressing somebody else in a lofty kind of way, you have some idea of what "Come Sleep! O Sleep" sounds like. Just compare the opening of the poem, which like Romeo's speech is full of praise: "Come Sleep! O Sleep, the certain knot of peace / The baiting place of wit, the balm of woe" (1-2).
If Sidney's poem is a little like his bro Shakespeare's play, it's also similar to something you might hear from a sportscaster on TV. (That's right. We said it.) Like, say, Al Michaels or Dick Vitale, the speaker keeps adding description after description after description: "The poor man's wealth, the prisoner's release / The indifferent judge between the high and low" (3-4). It's not until line 5 that we get another verb ("shield"). Compare this to something we just made up but that you could have heard during the 1996 NBA Finals: "Michael Jordan, the man, the myth, the legend, the greatest ever, Air Jordan..." It sounds a bit like: "Sleep, balm of woe, knot of peace, / baiting place of wit" (1-2).
The poem continues to add clauses in this way; there aren't a lot of verbs, now that we think of it. Just take a peek at lines 9-11: "Take thou of me smooth pillows, sweetest bed, / A chamber deaf to noise and blind to light, / A rosy garland and a weary head." We've put the verb in italics just so you can see what we mean.
Okay, so Sir Philip Shakespeare the sportscaster is narrating this poem—good to know. But what else can we say about the other neat little sound tricks going on here? Just check out the oh-so-clever internal rhyme ("sleep" and "peace" in line 1, "thine by right" in line 12), or the alliteration ("baiting […] balm" in line 2, "darts Despair […] doth" in line 6, just to name two). With these sonic flourishes, Sidney, like Shakespeare and our friend the sportscaster, makes the poem exciting and so all the more persuasive. To use the modern terminology, these techniques make the poem pop (see, we can't resist a good alliteration either) and subtly get across the excitement with which the speaker regards that all-star of all-stars: Sleep.