How does the concept of the labyrinth fit into “The Garden of the Forking Paths"?

Intriguing question! Borges's story is at its heart a labyrinthine enigma. We all know what a physical labyrinth is: it's basically a maze, a network of intricate passages, one of which leads to the center. The Greek mythological labyrinth (built by Daedalus to confine the Minotaur) was of a multicursal (or bifurcating) design. In other words, the labyrinth had branched passages that either converged with or diverged from other passages as they meandered along their paths. 
The concept of the labyrinth fits into the story in the sense that it epitomizes or symbolizes the bifurcating or multicursal nature of time in Borges's narrative. Here's a passage that explains this idea of time:

In the work of Ts'ui Pen, all possible outcomes occur; each one is the point of departure for other forkings. Sometimes the paths of this labyrinth converge: for example, you arrive at this house, but in one of the possible paths, you are my enemy; in another, my friend.

In the passage above, Stephen Albert, the Sinologist, explains to Yu Tsun the idea behind Ts'ui Pen's work. Ts'ui Pen's "indeterminate heap of contradictory drafts" appears on the surface to be the elements of a "chaotic novel." However, Stephen Albert hypothesizes that the drafts symbolize the bifurcating nature of time, which consists of "diverse futures" that "proliferate and fork." Ts'ui Pen's work (and Borges's as well) is an abstract conception of the physical labyrinth: all paths are possible, and each path can converge into or diverge from other paths. 
Anyone who has ever been trapped in a labyrinth will eventually experience some level of confusion or disequilibrium. Yet if we look at the labyrinth from above, the view is markedly different. Among the intricate and meandering passages, there is at least one that leads to the center. From above, we can see the purpose and design of the labyrinth, a privileged view that is denied the one trapped in it. Additionally, there may even be parallel paths leading to the center, a fact that will not be evident to anyone on ground level.
Borges makes this point clear in his story. Yu Tsun believes that his spy colleague has either been murdered or arrested. However, the "Editor's note" proclaims that Captain Richard Madden killed Runeberg in self-defense. This means that Runeberg must have attacked Madden, causing the latter to fire his weapon in self-defense. At this point, the reader may ask: Who is the "editor"? Is it Yu Tsun or Borges? Can both perspectives of Runeberg's fate be true? The labyrinth concept says yes, it can. Remember that the perspective of someone trapped in a labyrinth at ground level will differ from that of someone who has a panoramic view from above.
The labyrinth concept explains our dilemma in life. Hindsight is 20/20, as the saying goes, but how many of us have it in all the circumstances of our lives? To summarize, here are the two ways that the concept of the labyrinth fits into the story:
a) As a symbol of the multiple universe theory, where alternate universes demonstrate a multiplicity of possible outcomes.

I told myself that the duel had already begun and that I had won the first encounter by frustrating, even if for forty minutes, even if by a stroke of fate, the attack of my adversary. I argued that this slightest of victories foreshadowed a total victory. [I argued that the victory] was not so slight, since without this precious difference that the train schedule provided me I would be imprisoned, or dead....

In an alternate universe, Yu Tsun escapes capture and imprisonment. In the universe Yu Tsun is currently in, he's not so lucky. He must tell his story in prison, while he awaits execution.
b) As a symbol of the multiplicity of perspectives regarding the same situation/circumstance.

In the first, an army marches to a battle across a lonely mountain; the horror of the rocks and shadows makes the men undervalue their lives and they gain an easy victory. In the second, the same army traverses a palace where a great festival is taking place; the resplendent battle seems to them a continuation of the celebration and they win the victory.

In the above, Stephen Albert reads to Yu Tsun from Ts'ui Pen's work; they are essentially two versions of the same chapter. The perspectives are different, but the outcome is the same. 
https://www.borges.pitt.edu/sites/default/files/1808.pdf