This poem sets a happy scene. We have peaceful birds, pretty flowers, and a bright afternoon sun—all things to help us enjoy a fine spring day. We like to think of this as a kind of Bob Ross painting—you know, just 100 or so years before Bob picked up a paint brush.
We say more about this scene over in "Setting." For now, though, we'll just note that the sounds of the poem go along with the enticing content to add extra emphasis to certain lines and images.
Primarily, Browning uses a technique called alliteration to create such an emphasis, stringing beginning word sounds together to trip off the tongue. Check out all the B words, for example, in these lines:
That the lowest boughs and the brushwood sheaf
Round the elm-tree bole are in tiny leaf,
While the chaffinch sings on the orchard bough (5-7)
We also get repeated S words:
That's the wise thrush; he sings each song twice over, (14)
Line 14 throws some added assonance into the mix, echoing the long I sounds of "wise" and "twice" in the same line. Not too much later, we're treated to more alliteration, this time with F words:
The first fine careless rapture! (16)
All of these sonic effects create a kind of pleasant echo, as sounds echo and bounce off one another—much like birds chirping in a sunny spring meadow. Bob Ross has his happy trees; this poem has its happy sounds, each adding to the general impression of how great it would be to chilling in an English spring meadow.