Pilgrim's Progress is narrated not by Christian or Christiana, but by an unnamed narrator who relates a sequence of long, detailed dreams. The novel opens with a scant introduction to the narrator, setting up a frame story:
As I walk’d through the wilderness of this world, I lighted on a certain place where was a Den, and I laid me down in that place to sleep; and as I slept, I dreamed a Dream.
Interestingly, the narrator seems to be on a "pilgrimage" of their own, journeying through the metaphorical "wilderness" of the world. Though the novel doesn't draw this explicit connection, the narrator's journey seems to give a sufficient pretext for the type of dream he has when he naps in the den: he's traveling through the world's obstacles and dangers, so naturally, he dreams about the deeper, spiritual meanings such a journey can hold. Thus, the frame story helps readers anticipate what Bunyan wants them to gain spiritually from his allegory.
After this, readers seldom hear from the narrator. He sometimes transitions between events in the story by saying things like, "Then I saw in my dream...," and Part 1 simply ends with, "So I awoke, and behold it was a dream." The same pattern holds true in Part 2. The narrator intrudes very little into the action, then, giving a straightforward account of events without offering commentary or other interpretive remarks about what's happening to the characters. Besides the dream-as-frame-story providing Bunyan a handy way to set up a first-person omniscient narrator who's at a slight remove from the action, the dream can also be read as allowing for a layer of distance between Bunyan and the central allegory itself. That is, if it's just a dream, then readers shouldn't take his allegory too literally; it's simply a perspective on religious subjects and shouldn't be applied woodenly or as a prescription for every Christian's experience.
Finally, it's worth noting that dreams or visions are something of a literary trope in religious writings, with Dante Alighieri's narrative poem The Divine Comedy being the best-known example. In fact, the first volume of The Divine Comedy famously begins, "Midway this way of life we're bound upon, / I woke to find myself in a dark wood, / Where the right road was wholly lost and gone." Given that Dante has a vision and Bunyan's narrator has a dream—perhaps more acceptable to 17th-century Protestant readers, who were less inclined to mysticism than Dante's medieval Christian audience—the similarity is striking and probably not accidental.