Here are the first two paragraphs of the story in which the quoted sentences appear:
Knowing that Mrs. Mallard was afflicted with a heart trouble, great care was taken to break to her as gently as possible the news of her husband's death.
It was her sister Josephine who told her, in broken sentences; veiled hints that revealed in half concealing. Her husband's friend Richards was there, too, near her. It was he who had been in the newspaper office when intelligence of the railroad disaster was received, with Brently Mallard's name leading the list of 'killed.' He had only taken the time to assure himself of its truth by a second telegram, and had hastened to forestall any less careful, less tender friend in bearing the sad message.
The idea conveyed by "broken sentences" is of speech in which the message is expressed in halting or faltering words, so that the effect is as though the speaker were making several attempts to express the message but abandoning the line of thought partway through each time, or or as though the speaker were unable to complete the message through an excess of emotion.
So, in the story in question, instead of saying something like this:
"My dear sister, I am so terribly sorry to have to tell you this, but the train that your beloved husband was riding on this morning has gone off the tracks and crashed, and he is among the passengers who have lost their lives."
Josephine must have said something like this:
"My dear sister, I am so terribly—I hardly know where to begin... I must implore you, dear Louise—but please sit down first. We must be strong—and you must steel yourself... At a time like this... Oh, what a calamity! If only he had not been on that train—and on this day of all days! But what I mean to say, dear, is that your beloved husband, poor Brently—oh, it is too hard! But we must, as I say, be brave..."
These are broken sentences because, even though one can find complete sentences (such as "I hardly know where to begin" and "We must be strong") in the course of the quotation, the speaker is not assembling her words, sentences, and fragments into a coherent narrative.