It's not that Barzini set up the meeting. After all, Vito initiated the meeting in the first place. But Vito knew that someone other than Tattaglia was backing Sollozzo.

I think that Vito went into the meeting with two goals:

1. To get the other Families to agree not to deal drugs. Vito realizes this is futile after Don Zaluchi's speech.

and

2. To find out who was behind Sollozzo/Tattaglia. When Barzini took over the meeting and aired grievances, Vito knew that he was the secret adversary.