Mindfulness practice invites us to pay attention to the thoughts, feelings, and body sensations arising inside us when we feel agitated and distressed (or, of that matter, elated and invincible). Doing so, we spend a little more time attending inward, and a little less looking to people and situations as the source of our distress--and the target of our plan to fix things.
With less energy directed toward the "other" as a source of our discomfort and means to feel better, we are less likely to "judge" others.
Mindfulness is often referred to as non-judgmental awareness--which most succinctly can be understood to mean that we free ourselves from the weight of being judgmental when we notice our judgments, and pay attention to the thoughts, feelings, and sensations that accompany them.
Through the lens of mindfulness, the wise proscription "Judge Not Lest Ye Be Judged" takes on an added meaning. Not only is it a reminder not to judge others, it also speaks to the way the structure and function of our brains--and of our personality--may change as we develop greater mastery over our "judgmental" natures.
"Just not lest ye be judged" comes to mean in a very direct and personal way that as as we refrain from judging others--and become more naturally inclined in this direction--we will not feel judged by others.
Judging and being judged are interpretations. As we become less judgmental of others, we will be less likely to interpret the words and deeds of others as being judgmental. Even more, we will not feel the sting of "being judged."
Judge Not and Yee Will Not Feel Judged