The teaching of "being silent" refers to the nature of your mind; it is to invite a quiet stillness into your entire sense of being.
Being silent doesn't mean you can't talk or go about your daily business, it means no longer reacting to any of the egoic thoughts that arise within the mind—thoughts which only try to separate you from your natural presence of peace.
Being quiet within is the welcoming of mindful self-awareness—pure presence—which creates space around the mind’s thoughts so that you can observe them more clearly, without being personally attached to them.
During this silent witnessing of mental activity, it will become apparent how the mind projects itself onto every sense perception, which causes an illusion of division to appear between you and your experience of the world, which creates the appearance of many personal problems.
No longer be interested in the mind and its subjective noise, as it is this alone that is the cause of all unease, unrest, and dissatisfaction.
Understand that any reaction you have to anyone or anything can only come from the false self (the ego); it is never a reflection of the true Self.
Pure awareness—the Self—has no need to react; it accepts and allows whatever happens to appear within it. Awareness intuitively knows that nothing can affect it, as all phenomenal appearances within it are inherently empty. It is only the mind's attention onto a thing that creates a something out of what is naturally nothing.
To know your true self, you must be still and quiet—within—so that your natural awareness can shine onto everyone and everything, in every moment, while being undivided by the mind's stories, philosophies, judgements, beliefs, and mental concepts.
Awaken from the dream of mind.
Realize that you are its witnessing presence—an unbound space of infinite consciousness. Wake up from the false-reality which the ego projects onto the world—you are not it, nor are you any notion which it attempts to infer onto yourself, others, or the world.
Relax your grip on thought. Be quiet and allow it to take its natural course, without your emotional investment, effort, or involvement. Observe the mind’s activity so that your original nature becomes evident, which is the unshakeable foundation upon which every experience appears—the pure, undivided presence of absolute awareness.
Whatever you can observe is not you, and in your inner quietude, this will become apparent.
Stillness speaks. Be the quiet presence that listens.