In this context, Brene Brown is not talking about telling your story in the sense of a narrative. She is talking about the ability to be fully yourself and to allow others to see you fully. It is perhaps the most simultaneously frightening while also liberating display of vulnerability anyone can hope to accomplish because the world we have created for ourselves does, itself, simultaneously mock, but then also celebrate and admire those that do. It is these paradoxes that make being fully ourselves almost impossible to achieve.
Brene notes in her research that the common denominator among people who are able to live wholeheartedly (fully embracing who they are) is their ability to lean into joy. Joy is the most vulnerable of all the emotions because, in order to fully embrace it, you have to let go of shame and guilt; the fear of the judgment from others (and yourself); and be able to live in the present moment, entirely free of the fear of uncertainty. It's no easy task.
Like most things in life, the things worth fighting for are not easily won. Learning wholehearted living requires as much conscious effort and practice as you would expect to have to put into anything else that is difficult to do; and a courage that is as rare as it is misunderstood.