I've just posted a question and a few moments later, this message was displayed below it. I read about bounty and I understood what it means and my question is pretty specific. But why is it marked as "eligible for bounty" without any consent of my part?
Bounties can be offered by any user with sufficient rep on any question; consent is never involved. The only other condition is that the question isn't new. New questions don't need to be bountied, since they already show up in the New tab and the Active tab and so on (and thus get attention anyways).