"Peter has the biggest mouth, so it goes to be a mouth contest"

This quote by Josh Silver is likely referring to a circumstance in which Peter has actually been yapping and is being compared to others in regards to the size of his mouth. It could be translated as a joke, as if Peter is so talkative that he might win a contest for having the biggest mouth. It might also be translated as an obstacle, as if Josh is suggesting that Peter needs to enter a mouth contest to show that he has the most significant mouth. In either case, the quote indicates that Peter is extremely talkative and that his mouth is larger than those of the other people in the group.