In order to fully understand the symbolism of the open window of the title and the story, you must understand what sort of window it is. It is an old fashioned French window that is four or five feet wide and that extends from ceiling to floor. It has a sash that is pulled upward to stop overhead by way of weights and pulleys. It is meant to be walked through when open as doorway would be walked through. By its nature, it symbolizes the passage of people and events in and out of life.
In Saki's story, the open window symbolizes the dread Framton has of things passing in and out of his life without his having any control over what comes or goes. Consider, he is in the country visiting strangers against his will: his sister insisted upon his visits and arranged letters of introduction, which he might have disregarded or thrown out if he felt he were in control of the comings and goings of his own life; he even might have selected a different "rural retreat." He had a distinct feeling of unpleasantness about the visits with strangers, further emphasizing his dread of the uncontrolled comings and goings in his life:
"My sister was staying here, at the rectory, you know, some four years ago, and she gave me letters of introduction to some of the people here."
He made the last statement in a tone of distinct regret.
Other views on the symbolism include the idea that the open window ironically symbolizes deceit and deception, such as Vera's. It is ironic symbolism since open windows are more often thought to symbolize honesty. Another view is that it ironically symbolizes appearance versus reality since windows are meant to show the reality of the view opening from them. In contrast, Framton saw not the reality but the illusion of the appearance of reality: he saw three phantoms and a phantom dog but the phantoms he saw were real beings and contrasted with the appearance of reality. These explanations of symbolism seem weaker and less connected to the overall theme of the story than does the symbol of Framton's fear of uncontrolled comings and goings in his life.
In light of the titular (i.e., of the title) symbolism, the dominant theme might be stated as this: Mental instability comes from weakness of mind that fails to control who and what comes and goes in one's life.
To confirm this theme, had Framton controlled what he did in his rural retreat and what he did with those letters of introduction (out a train window might have been a good repository in his century), Vera would not have come into his life in the manner in which she did. Therefore the symbolism of the open window relates to the theme by making the open window the focal point of Framton's life story in the same way that it is the focal point of Vera's story.