They meet regularly to discuss local events, repairs, but mostly the matsuri (summer festival) and the bonenkai (the booze event of the end of the year). But they are also the ones who tell you who to support for elections. This year the local kumicho (head of kumi) came to give us leaflets for a city hall candidate, which surprised me because I know he dislikes the guy. But the boss of the "super-kumi" supports him so he gave lip-service to that. Knowing that this is the supported candidate by the local kumis will win a lot of local voters.
Another thing to understand is that seniority is very respected in Japan. I have seen a 63 yo man defer to a 65 yo because he was his "sempai". I don;t know if all the kumis are like that but ours is pretty much based on the assumption that the elder can boss around the younger.
That can explain why the conservative party is so well entrenched.
People polled in that article claim that they base their appreciation on managerial qualities and personal advantage, but from what I have seen, they get these opinions from their own social environment.