The first answer that often leaps to mind for many people is that this "becoming one" refers to the sexual relationship between husband and wife. True, and it's important to acknowledge that there is a deep understanding in the Bible that physical union between male and female produces a oneness that goes far beyond a momentary act. This is why "casual" sex is so terribly damaging. If we don't understand that this physical act produces deep spiritual and emotional bonds, we will damage ourselves very quickly. We take what God designed to be part of a lifelong relationship of intimacy and we make it as casual as sharing french fries. (Of course, there are some cultures in the world where you wouldn't share french fries with just anyone, either!) When God included a commandment about sexual purity in the Ten Commandments, he was letting us know that this is serious -- and if we treat it casually we're going to get hurt.
But "one flesh" refers to much more than just our sexual relationship.
One of the best examples I've ever seen of a "one flesh" marriage is a couple who have been married for many decades -- not many years, but many decades -- so they've had a long time to get used to each other. Some couples survive in marriage that long by distancing themselves from each other, but not these two. They have shared joy and grief, work and play and everything all their lives. I watched the two of them making breakfast once when Julie and I stayed with them. They had a narrow space to work in the kitchen. They maintained a conversation with the two of us while they worked. He was making orange juice, she was mixing pancake batter. I couldn't help but notice that while he went to the cupboard for a pitcher on his end of the kitchen, he also grabbed a mixing spoon from the drawer because he knew she would need it. When she retrieved eggs out of the fridge, she also got the frozen orange juice out of the freezer for him. Neither of them asked the other to do anything -- they were just very aware of what the other was doing, and they functioned like one four-armed person. It was almost as if they shared a mind and a body. Exactly. (See 1 Corinthians 7:4 and Ephesians 5:21.) This relationship took time and commitment to create. This couple spent years watching out for each other, paying attention to each other's needs, and learning how to serve one another.
Let me go a little further out on a limb with this "one flesh" idea. The Bible uses the word "flesh" in a couple different ways. Sometimes it just refers to the fact that our bodies are physical, like a T-bone steak is physical. (See for example Hebrews 2:14 or 1 John 4:2.) Other times, though, that word "flesh" refers to our old spiritual nature, our sinful self that fights against God's Spirit. (See for example Romans 8:12-13.) I wonder sometimes, is the Bible implying that husband and wife become, in a sense, one "flesh" so that my weaknesses, my sins, my rebellion against God become burdens my wife has to bear, and vice versa? This is a difficult thought and one that makes me uncomfortable, but I see its truth in action. If I carry unforgiveness around in my heart, it damages my wife, even if she is not part of the relationship where I'm holding a grudge. How much more does it hurt her if I am refusing to forgive her! When we have an area of our lives where we indulge our rebellious flesh and refuse to let Jesus have authority over us, husband and wife can cause each other some real problems. The most common areas where we experience this might include selfishness, unforgiveness, resentment, a need to control or manipulate, fear and anxiety, arrogance ... the list goes on. We bear one another's burdens (Galatians 6:2) as well as sharing one another's joys.