What is key to focus on in this quote is that it occurs after Mrs. Mallard has experienced the first shock of hearing about her husband's supposed death. This is part of a passage where the world outside Mrs. Mallard's room is described to us in all of its promise of spring and rebirth. Now that Mrs. Mallard has encountered the first shock of grief, she is able to think about what life would be like for her as a widow, and she finds that it is full of promise, hope and rebirth.
It is particularly significant in this quote that the "breath of rain" is described as being felt. Rain is of course something that helps growth and causes plants to flourish and bloom. The connotation helps us to focus on the way that Mrs. Mallard is now going to bloom as an individual now that she is free from her married state. This is one quote of many in this excellent short story that shows how nature parallels Mrs. Mallard's own feelings about being widowed.