I think the story goes like this. The Chinese came to North America, found nothing interesting and left. There was also a leadership strife in China at the time. The map was given or sold to the Europeans through existing trading channels. The implication is that Columbus knew there was land here before he set sail.
Here's a little bit of trivia. There is an ancient Chinese ship anchor found in Palos Verdes, California.
Personally, I think a lot of people knew land was there. Again, the Romans were here 2000 years ago. Perhaps Marco Polo told Kublia Khan about it.