They probably shook hands after a meeting, as one always does. That is not a "handshake deal". Airbnb probably had similarly positive meetings with dozens of investors. All of them probably talked about doing the deal soon.

He's known for being bold and writing checks on a first meeting. Why didn't he write a check to Airbnb after months of haggling? YC, or any other investor, was free to come along and close the deal at any moment. The way an investor claims a deal is with their checkbook (or a signed term sheet). He learned his lesson and realizes his mistake. Airbnb did nothing wrong.