It's definitely an unfinished and not highlighted feature right now (not mentioned in the tutorial).
As near as I can tell, you can see the inspiration symbol next to quests. Either you get them for completing that quest or maybe you have a chance to receive one depending on what you do. There is no feedback when you get one yet, and no count of them that I saw, just the symbol in the inventory screen. In D&D you only ever get 1, but I suspect they may change that to let you build them up (shrug).
My guess is that you're suppose to be able to use them on any dialog option that you fail, or possibly only a subset of them all (though that would feel inconsistent). I expect your experience with them is that you got some from a quest at some point and that prompted a possible reroll later. I think everyone starts with 1, but not sure.
TLDR: Feature is clearly unfinished and I'm sure it will either be made much more clear or removed later.