Dr. Jon Kabat-Zinn refers to non-judgmental awareness as to become aware of the constant stream of judging that our brain is doing in reaction to both the inner and outer experiences. It is not about completely getting rid of judgment. It is to learn to discern when its a time to support someone's insights or when it's necessary to give our own view and explanation.
Thanks to neuroplasticity we have the ability to change and alter our brain to create healthier pathways. Mindfulness practice can help rewire the brain so is no longer automatically reacting with anxiety, fear, anger, and judgment.
Remember that because our brain is constantly making a judgment, if left alone, the brain will automatically judge things and people as good/bad, right/wrong, fair/unfair, important/unimportant, you get the idea. This happens so fast that our experiences are automatically colored/tinted right when we get to them.
So the cultivation of non-judgmental awareness in mindfulness is about being aware, thus you can allow yourself to take a new perspective. The key here is once again AWARENESS.
Next time you have a human being in front of you, think about doing this practice:
Is the person in front of me just a person, or someone with their own history, triumphs, perceived failures, wanting the same things I do, to be understood and cared about?
When practicing mindfulness, it is important to recognize this judging quality of the mind when it appears and to intentionally assume the stance of an impartial witness by reminding yourself to just observe it. When you find the mind judging, you don’t have to stop it from doing that. All that is required is to be aware of it happening. No need to judge the judging and make matters even more complicated for yourself. ---Jon Kabat-Zinn