Lines 1-4 set the poem's scene and introduce its two main characters. These characters are never named; the speaker refers to them simply as "He" and "She." From the start, however, it's clear that they're a married couple who have been together for some time. By the end, the poem will reveal that they are the speaker's parents.
The couple is "Lying apart now, each in a separate bed." That is, they no longer share a marriage bed—though they do still seem to share a bedroom. The husband (the speaker's father) holds a "book" and "keep[s] the light on late," seemingly in order to read. In reality, he's not reading, as line 5 will reveal, so the book and the light may be a kind of avoidance tactic. Meanwhile, the wife (the speaker's mother) is "like a girl dreaming of childhood, / All men elsewhere." This simile suggests that she's taken a kind of refuge in nostalgia: that she's so unhappy with her current life and marriage, she's pining for a time when men were simply "elsewhere" and she didn't have to deal with them.
How can the reader know that these two are married and not in some other kind of relationship? The answer lies in the poem's title, which is a biblical allusion. The phrase "one flesh" appears a number of times in the Bible, always in contexts related to marriage. Here's Genesis 2:24 in the King James Version, for example:
Therefore shall a man leave his father and his mother, and shall cleave unto his wife: and they shall be one flesh.
The title is an indirect but strong hint that these two are man and wife. And in this context, the allusion is heavily ironic. "One flesh" suggests a passionate union of souls, as into one shared body—but this husband and wife don't even share a bed anymore! Though their marriage hasn't ended, it's clearly cooled off: they are both physically and emotionally "separate," and even their minds are roaming elsewhere.