Story of Your Life is narrated by Dr. Louise Banks on the night her daughter is conceived. Throughout the story, she narrates the events of her daughter's future life and recounts the arrival of a breed of aliens, referred to as "heptapods," on Earth. At the beginning of the story, she is approached by a government official, Colonel Weber, and a physicist, Dr. Gary Donnelly. Colonel Weber assigns Dr. Banks and Dr. Donnelly to a "looking glass": one of the communication devices that heptapods have placed at 112 different locations on Earth. Dr. Banks is assigned the task of learning about their language, while Dr. Donnelly is tasked with learning about their physics and mathematics.
Dr. Banks begins by recording spoken heptapod words, including "heptapod," "chair," and "yes." She uses audio recordings because she cannot make heptapod sounds herself, due to the fact that heptapods and humans do not have the same vocal tracts. She then moves on to studying heptapod writing, which she hopes will speed up her the learning process. She begins to compile heptapod words—both spoken and written versions of the same words. The heptapods use a logographic script, which means that they are not phonetically written. Dr. Banks and Dr. Donnelly nickname the two heptapods they communicate with "Flapper" and "Raspberry."
Dr. Banks discovers that their written and spoken languages are unrelated when she moves on to learning about heptapod verbs. In the spectrographs of the spoken language, Dr. Banks can differentiate the subject from the verb. However, in the written language, they are portrayed as a single logogram. Dr. Banks subsequently realizes that the heptapod script is not separated into words; instead, the words are brought together into a single image. Eventually, Dr. Banks realizes that the logograms should instead be called semagrams, because they are irreducible to spoken words. She demonstrates what this means to Dr. Donnelly by drawing a circle with a line through it on a board. This sign means "not allowed," but it does not correspond to any actual spoken words. She realizes that the heptapod written and spoken languages are completely different languages and decides to refer to them as "Heptapod A" (spoken) and "Heptapod B" (written).
The linguists decode the grammar of Heptapod A. They discover that heptapod speech contains free word order and center-embedding of clauses. Both of these features can be found in human languages. They also learn that the way in which semagrams are written (i.e. orientation or rotation) changes their meaning. Additionally, every single line that makes up the semagram must be read together with the rest of the semagram. Dr. Banks learns to write in Heptapod B, but she is not yet fluent.
As Dr. Banks makes advances in heptapod linguistics, the physicists are having less luck. Eventually, however, there is a breakthrough: the heptapods understand Fermat's Principle of Least Time. This principle states that a ray of light will take the fastest possible route to its destination. This requires for the ray to have an understanding of what the fastest possible route is.
Meanwhile, Dr. Banks asks to watch Flapper and Raspberry write semagrams, in order to determine if there is a preferred word order in writing. She discovers that when writing a semagram, they start with a single line that is integral to several different clauses in the sentence. This causes Dr. Banks to infer that "the heptapod had to know how the entire sentence would be laid out before it could write the very first stroke" (123). Later, Dr. Donnelly tells Dr. Banks that the heptapods do not see the universe as humans do. Humans have a causal interpretation of the universe (in which everything is part of a never-ending chain of cause and effect). In contrast, heptapods have a teleological view, in which they seem to know everything that happened and will happen at the same time.
As Dr. Banks becomes fluent in Heptapod B, she notices that it changes the way that she thinks. She wonders if the future is known, then if free will exists. In order for free will to truly exist, then someone cannot know what will happen in their future. However, Dr. Banks wonders, "what if the experience of knowing the future changed a person? What if it evokes a sense of urgency, a sense of obligation to act precisely as she knew she would?" (132). Dr. Banks buys a wooden bowl because she knows it will fall on her daughter's head in the future. This causes her to believe that free will still does exist, to an extent. As she buys the bowl, "the motion didn't feel like something I was forced to do. Instead it seemed just as urgent as my rushing to catch the bowl when it falls on you: an instinct that I felt right in following" (131). She realizes that the heptapods neither have true free will, nor are they completely bound to the future. Instead, their motives for acting simply coincide with the path of history. She reveals that now that she knows the future, "[she] would never act contrary to that future," even though she knows her daughter will die when she is 25 years old (137).
In the end, the heptapods leave after a gift exchange without any real explanation. Dr. Banks is there when they leave, and she knows what she will say because she can now see the future. The story closes in the present, where it is revealed that Banks's daughter's father is Dr. Donnelly. He asks her if she wants to make a baby.