What is the significance of the repeated references to rain throughout the story?

The rain signifies nature's way or function of cleansing itself. And, following this nuclear explosion and the decimation of the human population, there is the sense that nature is cleansing itself of the nuclear fallout and of the presence of humanity itself. In the fourth paragraph, the house mechanically sings, wishing the rain to go away, thus subtly symbolizing the conflict between technology and nature. 

The rain is used in other symbolic ways. At the...

The rain signifies nature's way or function of cleansing itself. And, following this nuclear explosion and the decimation of the human population, there is the sense that nature is cleansing itself of the nuclear fallout and of the presence of humanity itself. In the fourth paragraph, the house mechanically sings, wishing the rain to go away, thus subtly symbolizing the conflict between technology and nature. 

The rain is used in other symbolic ways. At the end of this paragraph, the rain hits the roof and the lonely echoes underscore the absence of the family, "And the rain tapped on the empty house, echoing." 

Following the descriptions of the white silhouettes of the family members, the narrator adds, "The gentle sprinkler rain filled the garden with falling light." Although this "rain" is a product of the house, it still does represent a cleansing. It follows the morbid description of what remains of the family members. And the sprinkler rain is "gentle," which is a clear contrast to the violence of the nuclear explosion. The "falling light" also suggests a peaceful sentiment; this is another contrast to the violence of the explosion. 

The rain in the children's nursery is "fresh," again suggesting something more natural and pure than war. Rain also comes up in Sara Teasdale's poem and in the title of this story. They are "soft" rains. This, again, is to show the contrast between the gentle, natural rain and the violent, synthetic nuclear bomb. In the end, the house even tries to save itself with its own mechanical rain. So, rain is once again used as a way to stop further destruction. It is nature's way of cleansing itself. It is also (usually) gentle and this creates a contrast with the violent result of a flawed humanity.