What is the significance of this quote from "The Story of an Hour"? "She could see the open square before her house the top of the trees that were all aquiver with new spring life. The delicious breath of rain was in the air. In the street below a peddler was crying his wares. The notes of a distant song which some one was singing reached her faintly, and countless sparrows were twittering in the eaves."

This quotation is significant because Mrs. Mallard has just learned of her husband's death in a train accident.  He was young and the accident was tragic.  (Of course, we later learn that the accident never happened at all, but Mrs. Mallard is unaware of this and believes that her husband is, in fact, gone.)  One would expect a typical wife's response to be grief, shock, and terrible sorrow and sadness.  However, this quotation shows that these are not Mrs. Mallard's feelings at all -- she is atypical in a very significant way.  Instead of dwelling on death and sadness, Mrs. Mallard seems to notice only signs of "new spring life": the scent of rain that has just ended is "delicious" and she hears the little birds "twittering" all around.  Further, she takes note of someone singing as well as a peddler calling out the names of the things he has for sale. 
Moreover, spring is often associated with new life as a result of trees and flowers blooming again and new animals being born after winter: Mrs. Mallard seems to come alive upon learning of her husband's death rather than being brought low because of it.  Her response, as seen in this quotation, is quite the opposite of what one would anticipate, and so much of the story has to do with why this is so.  We learn that, although Mr. Mallard was a loving husband, Mrs. Mallard felt trapped, repressed by her marriage, it seems, as a result of society's expectations for married women: she whispers the word "Free!" over and over, as though in disbelief that she has been set free of these expectations by her husband's passing.  She acknowledges that he loved her but that she had to "bend her will" to his in their relationship.  The new spring life that she notices outside -- the uncaged birds and so forth -- seems to suggest that she can now conceive of herself as "free" when she could not before.