Q. What does it mean when the Bible says that man and woman become "one flesh"? Does this refer to the children born of them, to a spiritual oneness, or what?
In its most basic, literal p'shat meaning, "one flesh" means the joining together of the male and female genital organs. It translates the Hebrew basar 'echad, where basar means 'body', 'flesh', 'skin' or the 'nakedness-self' but euphemistically the 'pudenda' (genitals), and 'echad means 'one'. (In a more extended sense it can also refer to one's biological 'kin' or 'relatives'). Basar is therefore to be exclusively understood in physical terms, and in this context, sexual ones. Basar 'echad therefore simply means the union of the sexes through intercourse. The farthest one can extend this concept would therefore be the physical oneness that obtained between Adam and Eve before they were separated. There is no direct sense in which Basar 'echad ever refers to children.
When it comes to expressing more than simply sexual union, i.e. spiritual union, the verb yada' is employed:
This word has a whole kaleidoscope of meanings including the literal, figurative, euphemistic and inferential. In its widest sense it means to have an intimate knowledge of a person but in this context it is again principally sexual. But whereas basar echad' might be said to only refer to the physical joining together, yada' implies a sexual union transcending intercourse itself to include an intimate 'knowing' by the parties of each other.
"Now Adam knew (yada') Eve his wife, and she conceived and bore Cain, and said, 'I have acquired a man from Yahweh'" (Genesis 4:1, NKJV)
Those espousing the monogamy-only position have argued that "one flesh" excludes the possibility of polygamy from the divine equation of marriage. But 1 Corinthians 6:16 shows that it is possible for a man to be one flesh with a wife and with a prostitute simultaneously and what makes the latter immoral is the fact that there is no marriage covenant.
It is worth noting that the prophet who penned Genesis 4:1 and 2:24 was himself married to two women - the Hebrew Zipporah and an unnamed Ethiopian (Cushite), traditionally from MeroŽ (Exodus 2:16-21; Numbers 12:1). When his sister Miriam criticised Moses about his second wife, Yahweh struck her with leprosy, indicating the blessedness of his polygamous union and Yahweh's vindication. Moses was in one-flesh union with both his wives since he had lawful sexual intercourse with them.
 One Flesh in The Truth About Biblical Marriage, Part 2, Chapter 11