Beyond the yield/price relationship, a good intuitive way to understand it is just this: these people control a substantial amount of money that could be essentially loaned to governments.
If they feel a particular policy is likely to lead to inflation or default, they may decide not to loan that country any more money. All else being equal, with a smaller supply of possible borrowers, the country will have to pay higher interest to fund a particular amount of debt.
Furthermore they may loudly publicly announce that they will no longer lend to that country, in which case other participants may be persuaded that they too should no longer lend at the going rate. What's more, this is somewhat self-fulfilling: as rates go up, the country will spend more money servicing its debt, and will in fact become a worse risk.
So I think the thing that gives them their "vigilante" nature is that governments worry they will round up a posse and things will run away.
As far as actual incentives, I would welcome more information but I think the main bond vigilante case is that they are basically long on the country but want it to tighten up its policy so their existing holdings don't decline.