Suppose 朝武さん made a mistake and felt embarrassed, so naturally he is trying to come up with excuses why he did it or why it happened. However he knows he will not be able to find a good excuse. However still he is trying to find an excuse because it is embarrassing or he cannot just let it go.
Another situation when 自分でもわかってはいるんだが can be used.
I have been trying to pass the bar exam to be a lawyer for 5 years now. My family, friends, colleagues all say it is time to give up and find another profession. I know it too that there is only slim chance I can pass even if I keep trying. However, I have invested so much time. I do not know how I can find another profession. Others may ridicule me if I give up.
In such a situation where odds are against you and you know it, then you would use the expression if you still try because you are so invested or obsessed or have some reason of not being able to stop.