For the sake of an unfamiliar reader, in the problem, we have a man standing by the buffers and an out of control trolley car. We're asked to choose between pulling a lever, which will divert the car to the buffers, thus killing the man but saving the people on board the car, or not pulling the lever, which results in the deaths of the trolley passengers but allows the man to live.
Choice one:
The out of control car is 'not your fault' but deliberately choosing to kill the man would be. You couldn't live with yourself knowing you'd been the cause of his death. You opt not to pull the lever.
Choice two:
That's crazy. None of this is your fault but it's your 'responsibility' to choose now. It's a simple utilitarian calculus - more lives = better. You pull the lever.
My problem with all this is that either response (and people split on the response) is nothing more than a reflection of the unconscious biases of the respondent. This is revealed when we discover that by changing the framing slightly - e.g. asking the question in another language (a different part of the brain is engaged = more utilitarians) or making the killing of the innocent man a direct act (you push him off an overpass = more 'not my faults') - changes the balance of responses in statistically significant ways.
The right answer, of course, is to toss a coin and do what it tells you.