I wasn't sure at first, but from watching the whole video it gets pretty clear; did you try that?
What it comes down to is that everyone gets to be a leader, in the before mentioned order. They will pick a team, which majority has to approve in order to go on the mission. If we don't approve the next person in line gets to choose a team. The people that are chosen to go on the mission get to choose whether the mission succeeds or fails, so for the non-spies it's important to always choose succeed, since we need to succeed 3 out of 5 and every win gets us a step closer. Only one person has to choose fail for the mission to fail so we have to always try to avoid getting a spy to come with us. If the mission fails it means there was a spy in the team. If it succeeds chances are that there wasn't a spy, or that a spy tries to gain trust to fail rounds later.
Edit: If you're a spy you have to try to gain trust and fail the missions. It's recommended to not fail missions with very few people in the team; that makes you suspicious. If it's one of the lasts missions you of course don't need this and can just hope for one of your fellow spies to get on the next mission if there is one. I might regret giving this explanation.. >_>