It goes like this:
Say your town has 30 Amish and 70 Mennonites. There's a Board that has 3 seats. Every year, all 3 seats go to the Mennonites, by a 70-30 vote.
Now they institute Cumulative Voting. Each voter gets to cast 3 votes. In this election 5 Mennonites and 2 Amish candidates are vying for the 3 spots. Now look at the possible outcomes...
Each Amish candidate gets 45 votes, as the 30 Amish with 3 votes each divide them perfectly between the two. There are 210 votes (70 x 3) for the 5 Mennonites. If they're perfectly spread out, they'd each get 42 (210 / 5). The board would have TWO Amish and only one Mennonite!
Of course, in the real world, votes don't distribute quite so evenly. Even so, the Mennonites could just ignore their bottom 2 candidates, and give the 210 to the other 3... at 70 each, they stiull get all 3 seats.
So what do the Amish do? They give ALL of their votes to ONE candidate. With 90 votes, it is IMPOSSIBLE for the Mennonites to keep them off of the board. With 210 votes, they can only top 90 votes for two candidates. Now the board is 2-1 Mennonite, but the minority Amish have a voice on the board... and pretty close to a proportaional prepresentation (33%, compared to 30% populatio and 28% of the candidates on the slate).
It isn't a perfect system, but it also isn't terrible or unfair. It assumes identity voting for many... but that isn't far from today's reality anyway. At least this way, larger minority groups have a very strong chance of gaining a voice on boards.
Why would the Mennonites run 5 candidates? Why not 3, and win all seats?
I understand your example; but is assumes poor planning and irrationality on the part of political parties.