Let’s take the trees first. The “under-natures” are the undersides of the leaves, which are “turned up” (made visible) by the wind, and since the undersides are paler than the upper surfaces of the leaves, this makes the whole tree appear paler, “blanching” it. The leaves “tremble” (vibrate) in the air currents, and the whole tree “dilates” (spreads out, widens) as its branches are bent by the wind.
Now, the words. The “under-natures” are the hidden meanings, nuances, connotations and allusions of the words, which are “turned up” (discovered, identified) by the reader. The words of the poems “tremble” (come alive) with “dilation”, that is, the expansion or enlargement of meaning produced by these discoveries.
How exactly this “blanches” the poem I am not entirely sure. The word “blanch” normally has negative connotations—to bleach, make pale, whitewash—but those senses do not seem right in this context. It seems that Browning uses “blanch” to mean “whiten” in the sense “to make morally or spiritually pure” (OED). Here’s another use of the word, from book VII:
No sun could die, nor yet be born, unseen
By dwellers at my villa: morn and eve
Were magnified before us in the pure
Illimitable space and pause of sky,
Intense as angels’ garments blanched with God,
Less blue than radiant.
The word “white” is also used elsewhere in the poem to mean “pure, good”, for example a portrait of the narrator’s deceased mother is described as “swan-like supernatural white life” (book I), and Marian’s “whiteness” (goodness) is “to be found / through all dark facts” (book VI).