Why do smart people tend to be quiet?

It's because they realize early in life that being too smart becomes a social obstacle so they downplay their intelligence to fit in. Gifted children are known to struggle socially in their teens due to this.

Smart people will often develop a habit of picking their battles because if they contribute to every conversation outshining everyone else with their intelligence, they are quickly seen as uppity and a know it all. For many smart people, it's hard to say anything at all without coming off as reminding everyone else of the disparity in intelligence. I believe there's some research that does show that being mentally superior is a social liability in a way that being physically superior is not. In other words, it's easier to be an athlete around non-athletes than it is to be a brainiac around non-brainiacs. Someone being too intelligent is instinctually felt as intimidating and stuck up.

For some smart people, they can't really hide their intelligence very well. It's built into how they speak. The way they structure their sentences. The cadence they use. Their vocabulary. In those cases, a smart person might choose to listen more and talk less so their intelligence is not dominating the social circle. So their intelligence isn't an elephant in the room.