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.
Being able to identify earlier rather than later that the conversation is being derailed to unproductive tracks is the task of the leader/facilitator, and interestingly it is very similar to the skill that mindfulness practice aims to do on a personal level. When practicing mindfulness meditation it is often highlighted that the brain generates thoughts that will try to grab your attention and take it away from whatever it is you're meditating on. These thoughts, at the moment they enter your awareness, tend to feel very important and urgent. They feel like if you won't follow this thread now it will not be there for you later. But then when you finish the meditation you will often find it hard to remember what that thought even was. What you are left with instead is a memory of a confused and dissatisfying meditation session. In order to overcome this natural stickiness of thoughts one must develop the 'outsider' perspective, the ability to take a step out of the thought and consider whether you want to follow that thread or if you want to let it go. Only from this outsider perspective do you have the power to consider, as objectively as possible, which choice is in your best interest, to follow the thread or let it go. As long as your attention is caught in the stickiness of the thought, the illusion of importance it creates is too powerful for objective considerations and it will be unlikely that you'd drop the thread fast enough to avoid derailment, even if it is the right thing to do. As a facilitator of meetings for your teams or organization it is your role to be aware of the moment when an irrelevant thread is threatening to hijack the meeting and derail it, and this is a skill that is essentially equivalent to the skill you'd develop and employ when practicing mindfulness meditation.
Another area where this skill comes into play for leaders is processes management, the set of rules according to which the team and company organize their efforts. In this area one has to maintain the "beginner mindset" or the "child mindset": Always questioning the established wisdom by asking 'why'. When following a process for a while, the understanding of why that process is there is slowly replaced with the habit of unquestioningly accepting its existence. Circumstances tend to always shift and change, though, and the circumstances that justified the process to begin with may alter or disappear over time. Many teams get bogged down with processes and rituals that were adopted or inherited in order to solve a set of circumstances that now no longer exist, or that are not a good match anymore when combined with newer processes and rituals. Yet they are still followed, unquestioned passively accepted as a reality that can't or shouldn't change.
For example, when my team got started we worked only with Story tickets. These stories contained a title and a detailed requirements for what needs to be done. Later on, we added a layer of epics as a process to tackle our difficulty organizing a messy backlog. We made the epics quite small so that each epic had only about 5 or 6 stories in it, and we'd finish about 4 or 5 of those epics per sprint. Just like the stories, each epic contained a title and detailed requirements, but we continued with writing detailed requirements in the stories as well, just like we did before. It then seemed to me that we are increasing our workload and violating the DRY principle by keeping the details in both the stories and the epics without getting any benefit out of it. If an epic already contains detailed requirements, the story can simply act as a pointer to a subset of the requirements without having to contain an actual copy of them. Everybody in the team was accustomed to the rule that stories must have the 'as... I want... So that' with acceptance criteria, so when I suggested that we try writing stories with just a title I was met with resistance and skepticism. But I managed to convince them to give it a shot, and when we stopped writing detailed description per story we indeed discovered that there was no loss of clarity, that our otherwise tedious refinement sessions became much more lightweight and that the price of creating stories was significantly reduced and therefore stories became much more atomic and focused.
It's no surprise I found it difficult to sell the idea of stories without a description; most of my teammates worked for years under the rule that each story must contained a specifically structured description. People tend to understand rules much better than they understand the reason for which the rule exists, so when you suggest that a rule could be changed because its raison d’être has changed you'd often find that it takes time for people to detach from the stickiness of the rule. The team tends to be committed to the way of working they are familiar with and tend to not question them. Rules, in a way like threads of thoughts, are sticky and have a natural air of importance and self justification. That's perfectly natural, but it is important that at least you, as the process leader, are not as affected by the stickiness as others. Someone has to initiate change by asking "why".
As a leader of facilitator, it is your role to continuously disentangle yourself from sticky matters, whether they are thoughts, discussions or processes, and question established ways, even if you were the one to originally establish them. Mindfulness meditation offers a direct exercise of the skill that helps you do that.