I haven’t seen this question raised. I can think of one reason, but I’m curious about other opinions.

Rand just comes to her door and says “It’s me”. Next thing we know, they’re off. She must have asked him how he knows, but we don’t get to see that.

My take is he told her about Min’s viewing of his father, as well as what his father told him when he took him to Nynaeve after the trolloc atttack. But this just proves he was born on Dragonmount. I don’t think the show ever said that was part of the foretelling. Is this enough to convince Moiraine?

He probably also told her that he thinks he channeled twice. But where does that leave him? He has no training. What can he possibly hope to accomplish at the Eye?

Edit: A number of people interpreted my question as a criticism the way it was handled in the show. I meant nothing of the sort, and I apologize for not making it clearer.

I expect a good show to leave things either to the imagination or analysis of the viewer (or both). I just believe that a discussion which fills in the gaps is a fun, useful thing to do. I get to see points that others picked up and I may have missed, or vice versa.