We are not very good at perceiving things as they are. We habitually add an interpretational layer to events and we unconsciously pay more attention to details that fit with the narrative we are currently invested in while overlooking or downplaying any evidence to the contrary. An insecure person is overly sensitive to other people's judgment of them, and an overly secure person tends to be oblivious to criticism.
Mindfulness, for the purpose of the article, is pretty much a synonym of Metacognition. It is the ability to consciously take a mental step back from what it is you're immersed in and experience it from a non-invested, (more) objective perspective.
To understand how this skill comes into play be for someone in a leadership or facilitator role, let's consider meetings. Infamous for being inefficient and coma inducing, as so often they get derailed, run beyond the allotted time and rarely achieve their intended goals. The reason this happens is not because the participants are too incompetent to achieve the goals or because there were not goals to begin with. In most cases the goals are clear and the participants have more than enough combined brain power to achieve them. But what often goes wrong is that the participants get immersed in the irrelevant things, such as following threads of discussions that lead them away from the intended goal or investing too much time in petty arguments. They follow these threads and arguments, wholly convinced, in the heat of the moment, that these are important and timely conversations, until the meeting runs out of time and the original purpose was barely discussed and no relevant decisions were made. Looking back on such a meeting one would conclude that it was a waste of time despite the fact that most people were following what at the time seemed like "important and timely conversations" to the best of their understanding. After enough of such experiences there's little surprise most of us develop an innate aversion to meetings as a wasteful endeavor. They are not, however, innately wasteful. Losing sight of the goal and chasing irrelevant threads is what makes meetings wasteful.