It’s easy to forget that you’ve got a first-person narrator in the driver’s seat here. Aside from the "I" in the first paragraph, we don’t really hear anything from this narrator. Let’s take a look at what little info we have regarding our narrator:
As long ago as 1860 it was the proper thing to be born at home. At present, so I am told, the high gods of medicine have decreed that the first cries of the young shall be uttered upon the anesthetic air of a hospital, preferably a fashionable one. So young Mr. and Mrs. Roger Button were fifty years ahead of style when they decided, one day in the summer of 1860, that their first baby should be born in a hospital. Whether this anachronism had any bearing upon the astonishing history I am about to set down will never be known.
I shall tell you what occurred, and let you judge for yourself. (1.1.1-2)
This introduction sets the tone of the story and the expectation of its readers. It's almost as if the reader is being told: this is going to be a fantastical tale, so brace yourself. Notice that Fitzgerald presents "Benjamin Button" as though it really happened; he’s passing the story on to you. It sounds as though this guy is elderly. When he refers to the way things are done today in a hospital, he adds, "So I am told," as though he’s not really up to speed on these new-fangled protocols. It’s not unreasonable for us to assume that that narrator is telling us these things first-hand; he witnessed the curious Benjamin Button case and now he’s describing it to you.