From the official episode guide:
"..FLASHBACK a final time, to David and Jessica's home, where Sawyer is about to close the deal and walk away with $300,000. But when David and Jessica's young son emerges from the bedroom, Sawyer can't bring himself to do it. He calls things off, inadvertently reveals his affair with Jessica and leaves the money behind.
Back on the island, Sawyer sleeps, recovering with a few stitches in his arm. He wakes up to find Kate sitting nearby with his letter. She explains to him that since she first read it, she's been feeling that something wasn't quite right. So she continued to read it ? over and over again, when it occurred to her that he wrote the letter ? and a 1976 Bicentennial postmark on the envelope confirms it. Sawyer admits that he was just an 8 year old kid when he wrote it. He explains that when he was nineteen, he got into trouble where he owed a lot of money to some very scary men, so he took the name of the man he was hunting and became a "confidence man" himself. But just when we think Sawyer is softening by opening up to Kate, he shuts down and chases her away. "