What does it mean, to observe self?
You have correctly stated the question. In this school, we do not speak of self-observation. When self is observing, observation is partial. By partial I mean that observation is divided between observer and observed and I also mean that observation is biased—the observer is injecting into observing innumerable judgments and evaluations based on past experiences and conditioning which the observer is largely unconscious of. Therefore, much is not observed.
Observing must be impartial to be effective. For this to be so, the would-be observer must be part of what is observed. A quality of attention must be accessed that does not require the mediation of an observer and is therefore not limited by partiality determing what is observed. It is the self in all its manifestations that must be observed. Thus, observation of self, not self-observation.
How is this to be done?
Always the ‘how’ question. First, you must be willing to give up being the observer and trust that an attention much greater than your own can perform the needed task. You must be willing to be seen. Essentially, you have to learn to release the need to be in control. Patience is also required. There are exercises for uncovering insight into the nature of impartial attention which can help you to learn to trust it without interfering. But the main thing is to learn to recognize the impulse to act from self and release it.