The poem packs a lot of information into its first four words. First, the use of apostrophe—the speaker directly addressing someone who isn't there—immediately signals the speaker's private longing and desire. The choice to use the word "thee," an informal version of "you," lends an increased intimacy from the very beginning of the poem. It also suggests a lack of convention, since "thee" was already an old-fashioned term in Browning's time. Finally, the clear caesura after "thee"—in the form of an exclamation point followed by an em dash—provides the reader with a sense of the speaker's passion: her admission seems to tumble forth in a burst of emotion.
On the other side of that caesura, the speaker describes not her lover, of whom she thinks, but rather her own thoughts themselves. Her thoughts, she claims using a simile, "twine and bud" around her lover like vines wrapping around a tree.
Here the poet uses consonance (/d/, /b/, and /th/ sounds) and assonance (long /i/ and /ee/ sounds) to create euphony. In other words, the lines are very pleasurable, musical:
I think of thee!—my thoughts do twine and bud
About thee, as wild vines, about a tree,
The euphony in these opening lines signals the pleasure that the speaker takes in thinking about her lover, and also imbues the thoughts themselves with a kind of lushness and vitality: the speaker is fantasizing, daydreaming, maybe even obsessing.
The poem's meter also plays a part in the musicality of these opening lines. The first line is in perfect iambic pentameter (meaning there are five feet, each foot comprised of an unstressed followed by a stressed syllable):
I think of thee!—my thoughts do twine and bud
This which might lead the reader to believe they are in for a conventional sonnet. However, the second line already disrupts this assumption, adding stressed beats where they don't belong:
About thee, as wild vines, about a tree,
Caesura, too, lends to the second line an almost physical sense of the vines coiling around the tree; without the presence of the commas, the line would still make sense but have a much breezier, open feel to it. The pauses created by the commas almost create a sense of restrictiveness, a foreshadowing that perhaps the vines—which are representative of the speaker's thoughts—are not entirely a good thing, despite the pleasure the speaker takes in them.