After the meeting with the heads of the 5 families, Vito says that he "didn't know until this day that it was Barzini all along." What happened, or what was said at the meeting that incriminated Barzini?

(I'm sure this has already been discussed, but I was unable to locate the thread using the search function. Sorry! lol)

Also, how does Vito know exactly how the traitors will come at Michael? Could it be that he knew Tessio so well, that he could predict how he would make his move? Or maybe Vito's just mad smart.

Recall that, at the beginning of the meeting, Vito says he wants to "thank Don Barzini for helping me to arrange this meeting." But Barzini isn't acting like Vito's Little Helper. He sits at the head of the table (and is the only guy smoking a cigarette, not a cigar like the "Moustache Petes"), is the first to reply after Vito's speech, and characterizes Vito's refusal to help Sollozzo and Tattaglia as "not the actions of a friend." And, as soon as Vito says he's willing to do anything to make peace, Barzini jumps in: "Then it is agreed: The traffic in drugs will be permitted and Don Corleone will give it protection in the East." Did you hear Vito or anyone else propose that? He showed his true colors, and Vito recognized them.
The only surprising thing here is that Vito didn't see Barzini's hand before the meeting:
Sollozzo was an importer and wholesaler. In order to establish a big retail drugs business in the US, he needed a powerful Mob partner to sanction his franchise and protect him from competitors, and to provide operational support: muscle, feet on the street, distribution and sales. He also needed money as well as police and political protection. Sollozzo, no dummy, figured Vito was too risk-averse to be his operational partner. But Vito might put up some money, and rent out his cops and judges, if he saw that some other Mob family was taking the operational risks.
I believe Sollozzo approached Barzini first. He knew Barzini was the up-and-coming NY Don, the guy most ambitious to supplant Vito as top Mob boss, and the one whose ambitions would most benefit from the money brought in by Sollozzo's drugs. But Barzini probably told him: "Look, if you figured out that I'm hot to get ahead of Vito Corleone and can use drugs money to buy the police and political power that keeps him on top now, then Vito himself already figured it out. If he sees that I'm your partner, he'll say no just to keep me in my place. Go to Tattaglia--he's a pimp, Vito has no respect for him and won't see him as a threat. I'll put in a good word for you with Tattaglia and be his silent partner." When Vito said no anyway, and Sollozzo proposed killing Vito, Barzini said he'd support Sollozzo and Tattaglia. What did he have to lose? If Vito was killed, he'd automatically become top Don. If the plot failed, Tattaglia and Sollozzo would take the heat. That's what happened (although Sonny regarded all the other families as enemies).
Vito himself said, after the meeting, "Tattaglia is a pimp. Alone he could never have outfought Santino." That was true before the meeting. So why didn't he know it until after the meeting? Perhaps because he was looking for final proof (Barzini's behavior at the meeting), just as Michael, later, demanded that Carlo confess, even though Michael knew he was guilty.
How did Vito know how "they" would come at Michael? Not difficult. Since he knew Barzini was behind his assassination attempt and the drugs business, it was easy for him to figure out what he told Michael: "Whoever comes to you with this Barzini meeting, he's the traitor."

So what you're saying, Turnbull, is that Vito's mad smart? Gotcha.