Time and space are important to literary realism insofar as they help the novelist to create a sense of actual cause-and-effect and, especially, a sense of social reality.
"The realists endeavored to accurately represent contemporary culture and people from all walks of life" (eNotes).
Realism, as a literary studies term, currently refers mainly to the idea of a rather mechanistic psychological reality drawn from theories of the mind that can be called "behaviorist." Realist works of literature examined the cause-and-effect relationship between social/economic circumstances and individual motivations and behaviors.
These novels posed the question, "How does a person's environment determine that person's behavior?"
"Even though neoclassical literature might have spotlighted the exploits of a single hero, it would not have been rooted in that character's psychology, but in the [realist] novel the exploration of individual consciousness and perception became the primary concern of representation" (eNotes).
The term realist is commonly mistaken to mean something along a parallel line to photo-realism in painting. This is not entirely accurate.
Moving away from "romance" and Romanticism, which was highly reliant on symbolism and melodrama, the realist novelists of Europe came to focus on presenting "real people" in real social circumstances. In these efforts to depict life as it is lived, writers rely on a realistic time-line and attempt to use settings that reflect material realities - as opposed to fictional, fantastic or symbolically oriented settings.
We should note that the term "realism" has become rather strict and concrete in its usage in literary studies, but remains more flexible outside of academic circles. When used more flexibly, "realism" refers to a text's ability to conjure a believable sense of place and action.
However, in its strictest sense character-as-a-function-of-setting is what the term attempts to describe. Thus, realistic uses of space and time (settings) contribute a successfully realistic depiction of the psychology of behavior.
Those who write in the Realist style try to portray things as they are, as they appear in real life. The goal of Realist art and literature is to be as close to reality as possible. This is why the novel was the primary genre of the original realists. This genre allowed the writer to describe things directly (as opposed to the figurative or flowery language of poetry).
Aside from the depictions of the psychology of a character, where the language and literature can get abstract and maybe even nonlinear (flashbacks, etc.), a Realist work describes things in Real time. So, a Realist novel would describe things the way we actually would experience them in time and space. Such would be the inclusion of how one event led to, or caused, another: 'this happened and then this happened.' Realists wrote realistic dialogue. Realists shared a common thread of portraying reality with Naturalist writers who focused on humanity's engagement with nature.
A good example of a Realist work that portrays events in Real time (as we normally experience it) is Stephen Crane's short story "The Open Boat." This story also qualifies as a Naturalist work. In the following quote, there is no overly poetic description, just things as they are:
At the top of another wave the correspondent did as he was told. This time his eyes found a small, still thing on the edge of the moving ocean. It was exactly like the point of a pin. It took an anxious eye to find a lighthouse so tiny.
In this next quote, note the plain objective description of 'this happened, and then this happened, etc.'
The oiler and the correspondent rowed the tiny boat. And they rowed. They sat together in the same seat, and each rowed an oar. Then the oiler took both oars; then the correspondent took both oars; then the oiler; then the correspondent. They rowed and they rowed.
Compare this straightforward style of using time and space with a Romantic or Surrealist style in which time and space might be bent, reversed, or even distorted.
Realism is all about the reality of life and the cold hard facts that one faces in their daily lives. Time and space were seen to have existed beyond the human mind, in a different world and level. Realists did not see time and space as something that would find itself similar to mysticism and fantasy. Realists were all about keeping everything real, hence the name. So time and space were treated the same way.
The importance of realists looking at time and space through their particular lens is that contributed to the advancement of knowledge about it. It was not looked as some kind of fantasy or dream, and so in during the realism period, discussions of time and space had peaked interest.
In a more simple sentence: It was important because it helped people realize what it really was and had people discover more about it.
A great example of this would be "An Occurrence at Owl Creek Bridge." I would hate to give away the ending, but read it and then watch the Youtube video of the Twilight Zone depiction of the story. You'll love it and have a grasp of how time is used. The resolution is a perfect example of how Realism grapples with Romanticism.