Responsibility is often seen as a burden, or as blame.  We use the word here as "the ability to respond" -- creative power.  To claim responsibility means to get curious about a situation, to learn from it, to look for the ways that we may be creating it or keeping it going, to discover our motivations, and to find ways of creating what we really want.  Responsibility often means deciding; the word decide comes from the Latin decidere, "to cut off".  By deciding what we want, we cut off other options.  Another sort of cutting away is the cutting away of denial and false hope, leaving only genuine power.  The prayer, "God grant me the courage to change what I can, the grace to accept what I can't, and the wisdom to know the difference," is speaking about this sort of responsibility.
The Basic Shift:  Getting Curious
The attitude of curiousity or wonder is where learning and the ability to creatively respond begins.  Focusing on a situation you have been struggling with, ask a series of questions beginning with, "Hmm, I wonder..."  How did I create this situation?  What do I really want?  How might I create what I want?  Create your own wonder questions, and see what responses appear in your consciousness or in your life.