In meditation, we observe ourselves -- it's boring necessarily. That's how you detach from discursive mind, by realizing how dull it is most of the time. But being dull does not diminish the value of sitting. The observer effect describes a phenomenon in physics where the thing watched changes its behavior. So by sitting and observing self, we change without 'doing' anything.
For the scientist of self, identification with thought decreases through observation, and in that gap between thought and self (created by just sitting and counting your breaths) transformation may occur. It's like a bath for the brain, a rinse of the mechanism that makes patterns, psychology, habits.
Watching discursive mind, returning to breath each time, you can see how no thought or sensation matters much because they change rapidly. Like a babbling brook, no drop counts particularly and the flow of the whole cannot be stopped. Observing, we learn the not specialness of our selves and our thoughts. And that is liberating.
Somehow this makes space for new things to happen when we are up and about -- new reactions or no reactions. What we discover during the down-time is the activity of the discursive mind. And all the chatter does make you wonder, who is this I?
It is a good and old question and to penetrate it means escape from the prison of self, which scientists and spiritualists alike agree is but a fiction of the mind. For example, Patanjali's ancient yoga sutras advised the reader to seek not to understand the sutras but instead ask who is seeking to understand, implying there is someone behind that I with whom you identify.
Is that someone our cosmic consciousness, the space where I am not me and you are not you but all are we? Perhaps. Maybe it's much less special than that. But in sitting and watching self we allow for that gap and thus free ourselves up for any possibility.