"My Grandmother's Love Letters," by Hart Crane, speaks to the past and the memories the speaker has of his grandmother.
She is gone, but memories, as well as mementos of her life, remain for the speaker. The night offers no visible stars (as it's raining), "but those of memory," and the speaker believes that there is plenty of room for memory as there is room for the soft rain that falls.
There is room enough, too, for his grandmother's love letters, so long stored away that they have become fragile—"brown and soft" enough to "melt as snow." There is great space for the speaker to recall his grandmother—and wonder about "Elizabeth," the person— apart from the one he loved—so much more than a mere memory. However, the steps into this "space" must be "gentle," especially if he is tempted to delve into unknown territory. He takes the time to consider the letters that will teach him more about this woman. (It is not hard to imagine someone wanting to learn about a loved one who is gone, especially in that when we are younger, we often don't know enough to ask questions while they are still with us.)
The lines you have mentioned are rhyming couplets. Each stands alone on a separate line—ended by a period rather than starting on the first line and continuing to the second. One source suggests that the speaker is having second thoughts, and that the periods are evidence that he is pausing. Does he really want to search out the secrets not of his grandmother, but of "Elizabeth," a woman he really never knew?
In these two lines, the "invisible white hair" literally calls up the image of his grandmother: but because it is invisible, the connection is one felt rather than seen. To know what the "hair" refers to, we need context, so we look at "It is all hung…" "It" may then refer to the decision the speaker has to make as to whether he is prepared to meet "Elizabeth"—the woman she was before: before he was born; before she was a grandmother; and, probably before she was a mother. The person she was when she received these letters has not existed for a very long time—life changes dramatically when we marry and have children…and grandchildren. What will he make of this unknown person who he is forever linked to? Perhaps, too, because he is a grandson first, and a curious adult second, the child within him is unsure if he wants to know who she used to be because he has loved her always as a grandmother. The decision is an important one; it hangs by a fragile and "invisible white hair." The rest of the couplet creates a simile, or a comparison: the hair trembles "like" the limbs of a tree spread out through the air. The word "trembles" provides a sense of uncertainty, or fear, or even anticipation—or perhaps all three, as the speaker ponders what step next to take.
It is all hung by an invisible white hair.
It trembles as birch limbs webbing the air.
To my mind, there is a decision waiting to be made about whether the speaker wants to discover a new woman in the person of his grandmother. He may actually see the move with anticipation—the birch limbs "web" through the air, lifting their arms to loftier places—toward the sun: and "higher planes" are often symbolic of increased knowledge. I do not believe his choice is necessarily between good or bad knowledge, but a "before" and "after" thing—for once he crosses into the world of "Elizabeth," he cannot go back: he will forever know of "Elizabeth" before their lives were joined.